Vintage Capital as an Origin of Inequalities∗

Andreas Hornstein† Per Krusell‡ Giovanni L. Violante§

August 2002

Abstract

Does capital-embodied technological change play an important role in shaping labormarket inequalities? This paper addresses the question in a model with vintage capitaland search/matching frictions where costly capital investment leads to large hetero-geneity in productivity among vacancies in equilibrium. The paper first demonstratesanalytically how both technology growth and institutional variables affect equilibriumwage inequality, income shares and unemployment. Next, it applies the model to aquantitative evaluation of capital as an origin of wage inequality: at the current rate ofembodied productivity growth a 10-year vintage differential in capital translates intoa 6% wage gap. The model also allows a U.S.–continental Europe comparison: anembodied technological acceleration interacted with different labor market institutionscan explain a significant part of the differential rise in unemployment and capital shareand some of the differential dynamics in wage inequality.

∗We thank Balthazar Manzano, Claudio Michelacci, Gilles Saint-Paul, Chris Pissarides, and FrancescoRicci for their comments and seminar participants at Bank of England, Bank of Italy, Bocconi, Bristol,ESSIM 2000 (Tarragona), Exeter, LSE, NBER Summer Institute 2002, NYU, Pompeu Fabra, Southamp-ton, Tilburg, Toulouse, UCLA. Krusell thanks the NSF for research support. Any opinions expressed arethose of the authors and do not necessarily reflect those of the Federal Reserve Bank of Richmond or theFederal Reserve System. For correspondence, our e-mail addresses are: andreas.hornstein@rich.frb.org,pekr@troi.cc.rochester.edu, and g.violante@ucl.ac.uk.

†Federal Reserve Bank of Richmond.‡University of Rochester, Institute for International Economic Studies, and CEPR.§University College London, Institute of Fiscal Studies, and CEPR.

0

1 Introduction

Recently, we have witnessed striking changes in the technology used in the workplace. The

“new economy” can arguably be characterized as follows: (i) technological improvements

seem to be intimately connected with the introduction of new capital goods and (ii) these

improvements proceed at a faster rate than before. Over the past two decades the produc-

tivity gains associated with new investment have represented the major source of growth in

U.S. output per capita (Jorgenson 2001).

The fact that new technologies are “embodied” in capital opens the possibility of large

discrepancies in worker productivity, both within and between firms. In this paper we

argue that unless labor markets are perfectly competitive–so that identical workers are

paid the same wage, independently of the capital they work with–rapid, capital-embodied

technological change can be an important determinant of wage inequality. In addition, to

the extent that there are frictions in matching capital to labor and that there are rents to be

divided between them, the rate of unemployment and the income shares–other important

aspects of inequality–can also be affected by technological progress.

Our analysis rests on a general-equilibrium framework with three building blocks: vintage

capital, a frictional labor market, and wage bargaining. To model capital-embodied techno-

logical change we use a vintage capital framework where machines/jobs are costly-to-create

units of capital of different ages, corresponding to technologies with different productivity

levels. To model employment inequalities, we operate in the tradition of Diamond/Mortensen

and Pissarides-style models, where an aggregate matching function determines the meeting

rate between unemployed workers and vacant jobs. To model wage inequality and the divi-

sion of income between labor and capital, we follow the standard approach in this literature

whereby wages and profits are endogenously determined through Nash bargaining within the

worker-firm pair. We show the existence and uniqueness of an equilibrium in our model and

analyze how inequalities in the labor market are determined as a function of the economy’s

primitives: technology, frictions, and institutions. In particular, we study the role of the

rate of technological change itself and its interaction with economic policy in the form of

government intervention in the labor market.

We use our theory of labor market inequality implied by capital-embodied technological

change to provide quantitative answers to two substantive questions. First, we use a cali-

1

brated version of the model to account for the contribution of vintage capital to observed

wage inequality: how much of “residual” wage inequality, that is, inequality that cannot

be attributed to observable characteristics of workers, might be due to differences in capi-

tal? Quantitative theory is useful here because we believe that it is very difficult to identify

relative machine quality in the data; data on the age of capital is difficult to link to wage

inequality, and firm or plant age are very crude and indirect measures; more disaggregated

data is simply not available.1

The other question we take on is perhaps more ambitious. Over the past thirty years la-

bor market outcomes in the United States and continental European countries have changed

substantially and in very different ways. In the United States wage inequality jumped to

the highest levels in the postwar period, the labor share of income declined slightly, and

the unemployment rate remained remarkably stable. In sharp contrast, in most of the large

continental European economies, the wage structure did not change much at all, while the

labor share fell substantially and unemployment increased steadily. Over the very same

period, impressive technological improvements embodied in new vintages of capital (espe-

cially in information and communication equipment and software) induced the adoption of

new production technologies across virtually every developed economy. Can these facts be

accounted for with our theory of capital-embodied technological change and labor market

frictions? We study, in particular, whether the interaction between this growth channel and

certain labor market institutions, whose strength differs between US and Europe, can explain

quantitatively the different evolution of the various dimensions of labor market inequalities.

THE FACTS:

In Table 1 we report some key numbers on unemployment rate, wage inequality, and

labor shares for several OECD countries at five-year intervals from 1965 to 1995. We are

particularly interested in the comparison between United States and continental European

countries (averaged in the row labelled Europe Average).2

In 1965 the unemployment rate in virtually every European country was lower than in

the United States. Thirty years later, the opposite was true: the U.S. unemployment rate

1Any systematic relation between wages and capital quality in the data would also be hard to interpret,since workers’ unobservable characteristics are likely to be correlated with the capital they are matched with(e.g., one might expect some degree of positive sorting).

2For completeness, we include data in Table 1 for the UK and Canada, whose behavior falls somewherebetween that of the United States and Europe.

2

rose by 1.7% from 1965-1995, whereas the average rise for European countries is 8.4%. The

labor share of aggregate income has declined only marginally in the United States, by 1.5%

from 1965-1995, while on average it fell by almost 6 points in Europe. Wage inequality,

measured by the percentage differential between the ninth and the first earnings deciles for

male workers, rose only slightly in Europe by 4% in the past 15 years, and it even declined

in some countries (Belgium, Germany, and Norway). The sharp surge of earnings inequality

in the United States is well documented, see Katz and Autor 1999, and the OECD data

confirm a rise of almost 30% since 1980. Interestingly, the European averages hide much less

cross-country variation than one would expect given the raw nature of the comparison. For

example, in 11 out of the 14 continental European countries, the increase in unemployment

rate has been larger than 6%, and in 9 out of 14 countries the decline in the labor share has

been greater than 5%.

THE QUALITATIVE ANALYSIS:

Aghion and Howitt (1994) and Mortensen and Pissarides (1998) pioneered the research

on the relation between embodied productivity growth and unemployment in a frictional

labor market.3 In their standard models new capital is always costless to buy and, as a

result, vacancies all consist of the newest capital. In contrast, the key new feature of our

model is the existence of vacancy heterogeneity, i.e., vacancies differ with respect to the

quality of the equipment on the job. This new feature is important for two reasons.

First, in the standard model the vintage structure is purely a frictional phenomenon:

when the capital is matched with a worker, it ages until a break-up results from the capital

becoming too obsolete relative to the worker’s outside option. As matching becomes more

and more instantaneous–as the friction is made weaker–separation occurs earlier and ear-

lier; in the limit, with no matching friction, all capital is new, so vintage effects are absent.4

Although our analysis has several features in common with these studies, we model capital

3Jovanovic (1998) investigates analytically the relation between embodied productivity growth and wageinequality in a competitive assignment model with a continuum of vintages of capital and of types of workers.Our introduction of frictions in the labor market allows a study of unemployment and induces a different wagedetermination mechanism with specific implications for wage inequality. Interestingly, some key mechanismsof the frictionless economy carry over to the frictional model, as will become clear below.

4Mortensen and Pissarides (1998) present also a model where firms can upgrade their capital withoutnecessarily inducing the destruction of the match. Because upgrading the existing machine is costly, whiledestroying the job and opening a vacancy with the new capital entails only the search costs, it remains truethat as the frictions disappear, so does the vintage structure. We return on the upgrading issue later in thepaper.

3

differently. We view capital as costly to buy, and once capital has been purchased, it is

natural to use it until it is so obsolete that the workers are more efficiently used elsewhere–

since they can alternatively work with newer capital. Thus, a unit of capital has a natural

life-cycle. Labor market frictions will make the life of capital longer because it is not costless

for a worker to find new capital to work with–she may have to go through an unproductive

period of unemployment. In contrast to the existing literature, in the frictionless version of

our model, capital is used for a strictly positive time period before being scrapped. In this

sense, our model is the most natural extension of the standard competitive vintage capital

growth model (Solow 1960) to an economy with labor market frictions.

Second, the presence of a nontrivial distribution of vacancies introduces new economic

forces in the standard model of equilibrium unemployment. First, the existence of a nonzero

outside option for the firm reduces the match surplus proportionally to the firm’s meeting

rate. Thus, changes in the embodied productivity growth rate, which have an impact on

the equilibrium meeting rates, will affect the surplus through this new channel. In addi-

tion, changes in the rate of technical progress will affect the equilibrium age distribution of

vacancies and, through this channel, the worker’s outside option of searching.

The main result of our qualitative analysis is that, notwithstanding the increased com-

plexity that this heterogeneity introduces, we show that it is possible to maintain analytical

tractability in characterizing the chief features of an equilibrium. In particular, we can repre-

sent the equilibrium of the economy with two curves (job creation curve and job destruction

curve) in the two-dimensional space defined by the age of capital at destruction and the

labor market tightness. The shifts of the two curves following a permanent rise in the rate

of embodied productivity are unambiguous, which allows us to describe qualitatively the

response of unemployment, inequality, and income shares. We show in particular that an

economy with generous unemployment benefits is more likely to respond to such a faster

productivity growth rate with a rise in unemployment duration, while a laissez-faire type

economy is more prone to respond through a reduction in the life-length of capital and more

job separations.

The intuition for this result is intimately related to the new features of our model: when

capital is costly, there exists a minimum life-length of the job required to fully recover the set-

up cost even in the absence of frictions. A U.S.-type economy with a minimal welfare state

has low labor costs and, hence, “bad” jobs with very old capital are still profitable, so that

4

the optimal scrapping age of capital is relatively high and far away from the technological

minimum. In contrast, in a European-type economy with munificent welfare payments, firms

are forced to scrap old capital earlier. An increase in the productivity of capital is in essence

an “obsolescence” shock to which firms would like to respond by shortening the life of capital

and adopting the new vintages more quickly. However, while this is possible in a U.S.-type

economy, such margin of adjustment is not fully available to European-type economies, whose

life of capital is already very close to the technological minimum. Since the scrapping age

cannot decline enough, firms need to be compensated through a different margin–a higher

meeting probability–which translates into longer unemployment durations for workers. This

mechanism improves the bargaining power of firms and allows them to push workers closer

to their outside option (which is constant across workers). The consequence is a larger fall in

the labor share of output and a smaller rise in wage inequality in European-type economies.

This qualitative analysis is one of the keys to deciphering the results of the quantitative

exercise.

THE QUANTITATIVE EXERCISES:

The quantitative importance of capital-embodied technological change for residual wage

inequality and unemployment has, as far as we know, not been studied before. The difference

between the labor market experiences of the United States and continental Europe, however,

has been the object of a quantitative analysis in a number of papers.5 The divergent behavior

of the two economies is explained in these papers through the interaction between different

labor market institutions across regions and a common structural shock to the economic

environment. In our view, the existing literature does not offer a satisfactory way to link the

fundamental driving force behind the changes in the labor market to independent observable

data. As a consequence, any calibration attempt matches one of the crucial elements of

interest, such as the rise in inequality or the changes in income shares, by construction. We

take the view that unemployment, inequality, and changes in the labor income share are

of great importance and have to be explained jointly: they are dimensions along which the

model should be evaluated rather than calibrated. An important advantage of our model

is that the unique source of the shock is capital-embodied productivity, and the parameter

regulating the speed of capital-embodied technological change can be measured through

independent data–the change in the quality-adjusted relative price of equipment–as is done

5We summarize this literature in section 5.4.

5

in a number of previous papers that have applied this information to growth accounting and

analyses of the labor market.6

The model suggests that vintage capital has a significant impact on wage inequality,

although the implied level of wage inequality is small compared to the data in Table 1. This

result is not surprising, as the only source of wage differentials in our economy is vintage

capital within an ex ante equal set of workers. Because of the lack of detailed employer-

employee matched data where one could sharply distinguish the role of workers’ individual

characteristics from the role of firms’ characteristics in wage determination, we are not aware

of any direct empirical estimate of the effect of differences in the vintage of capital on wage

differentials. We can, however, use our calibrated model to give an answer to this question:

we find that in a U.S.-type economy a difference of ten years in the vintage of capital used

by the firm generates wage differentials around 6%. We argue that this represents about one

fourth of residual wage inequality for ex-ante equal workers in the United States.

The quantitative U.S.—Europe exercise consists of an acceleration in the rate of embod-

ied productivity growth in economies that differ according to the generosity of their welfare

benefits and the strictness of employment protection legislation. The main result of our

quantitative exercise is that the model is successful in generating the observed differential

rise in unemployment and in the capital share between the United States and Europe. A

permanent rise in the rate of capital-embodied productivity growth of 2 percentage points

increases unemployment rate by less than 1 point in the U.S.-type economy and by over 8

points in the European-type economy, with all the increase taking place along the unem-

ployment duration margin, as in the data. The labor share falls by over 6 points in both

economies, but once we introduce a firing tax to capture variations in the degree of employ-

ment protection, the model generates a stronger fall in the labor share (by circa 3 points) in

European-type economies with stricter firing restrictions.

Finally, the numerical simulations show that our model with vacancy heterogeneity dis-

plays a quantitative amount of technology-policy complementarity much larger than that of

the standard Aghion-Howitt/Mortensen-Pissarides framework. We believe this complemen-

tarity helps in explaining the data.

The remainder of the paper is organized as follows. In Section 2, we start our analysis

6See Gordon (1990), Hornstein and Krusell (1996), Greenwood, Hercowitz, and Krusell (1997), Greenwoodand Yorukoglu (1997), Krusell, Ohanian, Rıos-Rull, and Violante (2000), and Cummins and Violante (2002),among others.

6

with the frictionless environment, where workers are all paid the same wage and all are

employed. In Section 3 we move to the frictional environment with heterogeneous vacancies,

solve the model, and prove the existence and uniqueness of equilibrium. In Section 4 we

characterize how equilibrium inequalities in employment, wages, and income shares respond

qualitatively to a change in the speed of embodied technology, and we also study the role

of different labor market institutions in an attempt to explain the distinct labor market

performances of the United States and Europe. Section 5 presents the calibration of the

model and the results of our quantitative exercises and discusses the related literature in

detail. Section 5.5 compares our model with the standard matching model. Finally, Section

6 concludes the paper.

2 The frictionless economy

Time is continuous. The economy is populated by a stationary measure 1 of workers who

are all alike, live forever, are risk-neutral, and discount the future at rate r. Technological

progress is embodied in capital, and the productive capacity of new vintage machines grows

at the rate γ > 0. A firm (or job, or production unit) can be created through an initial

investment expenditure I(t), and the cost of new vintage machines also grows at the rate γ.

Firms can freely enter the market upon payment of the initial installation cost. At time t,

firms can choose whether to purchase the newest vintage machine or a machine of any older

existing vintage: newer vintages are relatively more expensive to set up, but they are also

relatively more productive.

A firm is productive only when paired with a worker. There is no physical depreciation of

machines and production of a firm remains constant through its lifetime. There is, however,

economic depreciation. Older firms produce relatively less than newer firms because of

embodied technological change, and firms with old enough capital will voluntarily exit the

market.

In order to make the model stationary, we normalize all variables and define output

relative to the newest production unit. The normalized cost of a new production unit is

then constant at I, and the normalized output of a production unit of age a which is paired

with a worker is e−γa. We will focus on the steady state of the normalized economy, which

corresponds to a balanced growth path of the actual economy. Finally, we will assume that

r > γ to guarantee the boundedness of infinite sums.

7

We start by describing the competitive equilibrium for the frictionless economy. In the

steady state the wage rate also grows at the rate γ and the normalized wage w ≤ 1, nowmeasured relative to the output of the newest vintage, is constant. Consider a price-taker

firm that plans to set up a new vintage machine. The firm optimally chooses the exit age a

that maximizes the present value of machine lifetime profits

maxa

Z a

0

e−ra(1− weγa)da ≡ Π(w),

where Π is the profit function. Since flow profits are monotonically declining and eventually

become negative, there is a unique exit age for new vintages. Profit maximization leads to

the condition

w = e−γa, (1)

stating that the price of labor has to equal the productivity of the oldest machine, which is

also the marginal productivity of labor. The higher the wage, the shorter the life-length of

capital since (normalized) profits per period fall and thus reach zero sooner.

We next argue that profit-maximizing firms always choose the newest capital vintage.

Suppose the labor required to operate new vintage machines was also increasing in the

quality of machines at rate γ over time. Then firms would be indifferent between the newest

and any older technology: an older vintage would simply scale down costs–both for the

machine and wage expenses–and revenues by the same amount, leaving profits unchanged,

and the time in operation would remain at the same level as that for new firms. The labor

requirement, however, is not increasing over time, which is why new technologies are better;

in fact, technological change is labor-augmenting here in the sense that it allows one worker

to work with more and more efficiency units of capital over time by using newer and newer

equipment. Thus, a firm choosing to invest in old capital would, once in operation, generate

lower profits per period, and it would operate for a shorter period of time (since the time

at which the wage equals the total product is reached sooner) than if it chose the newest

capital. The lower cost of the old machine would compensate these losses only partially.7

Free entry of firms requires that in equilibrium I = Π. This is the key condition that

determines exit age a, and hence wages. Using the profit-maximization condition (1), the

free entry condition can be written as

I =

Z a

0

e−ra£1− e−γ(a−a)

¤da. (2)

7This argument is easy to verify mathematically, so we omit its proof in the text.

8

Equation (2) allows us to discuss existence and uniqueness of the equilibrium as well as

comparative statics. It is straightforward to solve for efficient allocations and show that

a stationary solution to the planner’s problem reproduces the competitive allocations (see

Appendix A.1).

The right-hand side of the equilibrium condition (2) is strictly increasing in the exit age

a for two reasons. First, in an equilibrium with older firms, the relative productivity of the

marginal operating firm is lower and therefore wages have to be lower and profits higher.

Second, a longer machine life increases the duration for which profits are accumulated. The

right-hand side of (2) increases from 0 to 1/r as a goes from 0 to infinity. Taken together,

these facts mean that there exists a unique steady state exit age aCE whenever I < 1/r.

This condition is natural: unless you can recover the initial capital investment at zero wages

using an infinite lifetime (R∞0

e−rada = 1/r being the net profit from such an operation),

it is not profitable to start any firm. With a unit mass of workers, all employed, the firm

distribution is uniform with density 1/aCE, which is also the measure of entrant firms ef .

Turning to comparative statics, we note that a larger interest rate r decreases present-

value profits, thus lowering entry and increasing the life span of the machine. Conversely,

an increase in the cost of a new machine I will raise the life span: fewer machines enter and

they stay in operation longer to recover the fixed cost. An increased growth rate of capital-

embodied technological change γ must decrease the life span of machines and increase the

number of firms that enter at each point in time. Formally, the right-hand side of equation

(2) is increasing in the growth rate γ: the higher the growth rate, the lower the relative

productivity of the least productive firm, and therefore the lower the cost of hiring labor

must be. Faster growth therefore means higher profits, implying an increase in entry at

the expense of older machines that are forced to exit earlier. Thus, in the competitive

economy when technological change accelerates, the rate of job turnover in the economy

rises and, as a consequence of the decline in the wage rate, the labor share of aggregate

income ωCE = γ/³eγa

CE − 1´falls.

Although the prediction on the income shares qualitatively matches the facts of Table

1, it is worth remarking that the environment without frictions displays neither wage nor

employment inequality, so it cannot serve as a tool to analyze the facts we described. For

this reason, we now turn our attention to an environment with matching frictions.

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3 The economy with matching frictions

In this section, we consider a slightly different economy. The demographics and the tech-

nological side of the model are unchanged, but the structure of the labor market is new.

The labor market is no longer perfectly competitive: it is frictional. The matching process

between workers and production units is random and takes place in one pool comprising

all workers and all vacant firms; vacant firms are distinguished by the age of their capital.

Throughout, and for tractability, we will focus on steady-state analysis; thus, the notation

presumes no time-dependence. In particular, all distributions are stationary over time.

The nature of the firm’s decision process–buy a piece of capital, then match with a

worker, and finally exit when the capital is so old that it no longer generates positive profit

flows–remains the same as in the frictionless economy. In particular, firms in this economy

will also choose to buy the newest form of capital when entering. Due to the matching

frictions, some firms will also become idle, but idle firms have no option but to wait until

they meet a worker.8

The rate at which a worker meets a firm with capital of age a is λw(a) and the rate at

which she meets any firm is λw ≡R a0λw(a)da, where a is the job-destruction age. A firm

meets a worker at the rate λf . Let ν(a) denote the measure of vacant firms of age a. We

assume that the number of matches in any moment is determined by a constant returns

to scale matching function m(v, u), where v ≡ R a0ν(a)da is the total number of vacancies

and u is the total number of unemployed workers. We also assume that m(v, u) is strictly

increasing in both arguments and satisfies some standard regularity conditions.9 Using the

notation θ = v/u to denote labor market tightness, we then have that

λw(a) =ν(a)

vm(θ, 1), (3)

λf =m (θ, 1)

θ. (4)

8One can easily allow for an “upgrading” decision: in any given period, with some probability the firmhas an opportunity to upgrade the machine to the newest capital and keep the worker at some cost. In theinterest of keeping the model tractable for our qualitative analysis we abstract from this feature here.

9In particular,

m(0, u) = m(v, 0) = 0,

limu→∞mu(v, u) = lim

v→∞mv(v, u) = 0,

limu→0

mu(v, u) = limv→0

mv(v, u) = +∞.

10

The expression for the meeting probability in (4) provides a one-to-one (strictly decreasing)

mapping between λf and θ. Thereafter, when we discuss changes in λf , we imagine changes

in θ.

We assume that matches dissolve exogenously at the rate δ: upon dissolution, the worker

and the firm are thrown into the pool of searchers.10 Searching is costless: it only takes time.

When unemployed, the worker receives a welfare payment b. The measure of matches with

an a firm and a worker is denoted µ(a) and total employment µ.

Values for the market participants are J(a) and W (a) for matched firms and workers,

respectively, V (a) for vacant firms, and U for unemployed workers. Let w(a) denote the

wage paid to a worker from an a firm. The values solve the following differential equation

system, which summarizes the flow payoffs of workers and firms:

(r − γ)V (a) = max{λf [J(a)− V (a)] + V 0(a), 0} (5)

(r − γ)J(a) = max{e−γa − w(a)− δ [J(a)− V (a)] + J 0(a), (r − γ)V (a)} (6)

(r − γ)U = b+

Z a

0

λw(a) [W (a)− U ] da (7)

(r − γ)W (a) = max{w(a)− δ [W (a)− U ] +W 0(a), (r − γ)U}. (8)

The derivatives of the value functions with respect to a will be negative and are flow losses

due to the aging of capital.11

In the presence of frictions, a bilateral monopoly problem between the firm and the worker

arises, and thus wages are not competitive. As is standard in the literature, we choose a

Nash bargaining solution for wages. With outside options as in the above equations, the

wage is such that at every instant a fraction β of the total surplus S (a) of a type a match

goes to the worker and a fraction 1− β goes to the firm:

S (a) ≡ J (a) +W (a)− V (a)− U (9)

W (a) = U + βS (a) and J(a) = V (a) + (1− β)S (a) . (10)

The explicit solution for the wage is discussed in Section 4.2. Finally, we require that

V (0) = I so that there is no profitable entry by firms with new capital in equilibrium.

10We omitted this event from the description of the competitive equilibrium because, without frictions,it is immaterial to the firm whether the match dissolves exogenously or not as the worker can be replacedinstantaneously at no cost.11In Appendix A.2 we describe a typical derivation of the differential equations above.

11

3.1 Solving the matching model

We characterize the equilibrium of the matching model in terms of two variables: the rate

at which vacant firms meet workers and the exit age: (a, λf). The two variables are jointly

determined by two key conditions. The first condition, labelled the job destruction condition,

expresses the indifference between carrying on and separating for a match with capital of age

a. The second condition, labelled the job creation condition, expresses the indifference for

outside firms between creating a vacancy with the newest vintage and not entering. In the

next main section, Section 3.2, we then demonstrate that a solution to these two equations

exists and is unique.

In Section 3.1.1 we first derive closed-form solutions of the system of equations (5)—

(10) that define the value/surplus functions. The specific solution for the surplus function

depends on the pair (a, λf) and the unemployment value U . In Section 3.1.2 we apply the

results of Section 3.1.1 to the optimal separation decision and derive the job destruction

condition. The optimal separation decision does depend on the pair (a, λf) and the rates

λw (a) at which unemployed workers are matched with firms. In Section 3.1.3 we apply the

results of Section 3.1.1 to the free entry requirement and derive the job creation condition

which depends only on the pair (a, λf). In Section 3.1.4 we derive the rates λw (a) at which

unemployed workers are matched with firms in terms of the pair (a, λf).

3.1.1 The surplus function

In this class of models all decisions are surplus-maximizing. Thus, it is useful to start by

stating the (flow version of the) surplus equation. Using (9) this equation can be described

by

(r − γ)S(a) = max{e−γa − δS(a)− λf(1− β)S(a)− (r − γ)U + S0(a), 0}. (11)

This asset-pricing-like equation is obtained by combining equations (5)-(10): the return on

surplus on the left-hand side equals the flow gain on the right-hand side, where the flow

gain is the maximum of zero and the flow difference between total inside minus total outside

values. The inside value flows include (i) a production flow e−γa, (ii) a flow loss due to the

probability of a separation of the match δS(a), and (iii) changes in the value for the matched

parties, J 0(a) +W 0(a). The outside option flows are (i) the flow gain from the chance that

a vacant firm matches λf(1− β)S(a), (ii) the change in the value for the vacant firm V 0(a),

and (iii) the flow value of unemployment (r − γ)U .

12

The solution of the first-order linear differential equation (11) is the function

S(a) =

Z a

a

e−(r+δ+(1−β)λf )(a−a)£e−γa − eγ(a−a)(r − γ)U

¤da, (12)

where we have used the boundary condition associated with the fact that the surplus-

maximizing decision is to keep the match alive until an age a such that S(a) = 0. For lower

a’s the match will have strictly positive surplus, and for values of a above a the surplus

will be equal to zero. Straightforward integration of the right-hand side in (12) and further

differentiation shows that, over the range [0, a), the function S(a) is strictly decreasing and

convex; moreover, S(a) will approach 0 in such a manner that S0(a) is defined and equals

zero. Intuitively, the surplus is decreasing in age a for two reasons: first, the time-horizon

over which the flow surplus accrues to the pair shortens with a; second, the outside option

of the worker rises over time at rate γ — the pace of productivity growth of the new vacant

jobs — while output is fixed.

Equation (12) contains a non-standard term due to the vacancy heterogeneity: the

nonzero firm’s outside option of remaining vacant with its machine reduces the surplus by

increasing the “effective” discount factor through the term (1− β)λf . Everything else being

equal, the quasi-rents in the match are decreasing as the bargaining power of the firm or its

meeting rate is increasing.

3.1.2 The separation decision

The optimal separation rule S (a) = 0 together with equation (12) implies that the exit age

a satisfies

e−γa = (r − γ)U, (13)

for a given value of unemployment U . The idea is simple: firms with old enough capital

shut down because workers are too expensive, since the average productivity of vacancies

and, therefore, the workers’ outside option of searching, is growing at the rate of the leading

edge technology. Note that this equation resembles the profit-maximization condition in the

frictionless economy, with the worker’s flow outside option, (r− γ)U , playing the role of the

competitive wage rate.12

12In fact, later we show that the lowest wage paid in the economy (on machines of age a) exactly equalsthe flow value of unemployment.

13

We can now rewrite the surplus function (12) in terms of the two endogenous variables

(a, λf) only, by substituting for (r − γ)U from (13):

S(a; a, λf) =

Z a

a

e−(r+δ+(1−β)λf )(a−a)£e−γa − eγ(a−a−a)

¤da. (14)

In this equation, and occasionally below, we use a notation of values (the surplus in this

case) that shows an explicit dependence of a and λf . From (14) it is immediately clear that

S(a; a, λf) is strictly increasing in a and decreasing in λf . A longer life-span of capital a

increases the surplus at each age for two reasons. First, it increases the surplus flow because

it lowers the flow value of the worker’s outside option, (r − γ)U = e−γa. Second, it increases

the duration for which a match receives a positive surplus flow. A higher rate at which firms

meet workers, λf , reduces the surplus because it increases the outside option for a firm: a

vacant firm meets workers at a higher rate.

The optimal separation (or job destruction) condition (13) requires that the lowest output

in operation be equal to the flow value of unemployment. Using (7) and (10) we obtain

e−γa = b+ β

Z a

0

λw(a; a, λf)S(a; a, λf)da, (JD)

which is an equation in the two unknowns (a, λf) and the rates λw (a) at which unemployed

workers are matched with firms. In Section (3.1.4) below, we explain how the two endogenous

variables determine the workers’ meeting rates.

3.1.3 The free-entry condition

We define the value of a vacancy of age a using the new expression (14) for the surplus of

a match S(a; a, λf) together with (10). The differential equation for a vacant firm (5) then

implies that the net-present-value of a vacant firm equals

λf(1− β)

Z a

a

e−(r−γ)(a−a)S (a; a, λf) da, (15)

where a equals the age at which the vacant firm exits. Since vacant firms do not incur in

any direct search cost, they will exit the market at an age such that this expression equals 0,

from which it follows immediately that a = a. Since in equilibrium there are no profits from

entry, we must have that V (0; a, λf) = I, and we thus have the free-entry (or job creation)

condition, which becomes

I = λf(1− β)

Z a

0

e−(r−γ)aS(a; a, λf)da. (JC)

14

This condition requires that the cost of creating a new job I equals the value of a vacant firm

at age zero, which is the expected present value of the profits it will generate –a share (1− β)

of the discounted future surpluses produced by a match occurring at the instantaneous rate

λf . The job creation condition is the second equation in the two unknowns (a, λf).

3.1.4 The stationary distributions and measures

We now complete the characterization of the equilibrium and derive explicit expressions for

the matching probabilities in terms of the endogenous variables (a, λf). The probabilities

λw(a) depend on the steady-state distributions of vacant firms. The inflow of new firms is

ν(0): new firms acquire the new capital and proceed to the vacancy pool. Thereafter, these

firms transit stochastically back and forth between vacancy and match, and they exit at

a = a, whether vacant or matched (after matched, a firm can always become vacant at age

a < a at rate δ). This means that ν(a) + µ(a) = ν(0) for all a ∈ [0, a). The functions ν(a)and µ(a) jump down to 0 discontinuously at a. For a ∈ [0, a), the evolution of µ(a) thereforefollows

µ(a) = −δµ(a) + λfν(a) = λfν(0)− (δ + λf)µ(a). (16)

Exogenous separations δµ(a) reduce employment, and vacancies being filled λfν(a) increases

employment.13 It is easy to demonstrate that

µ(a)

µ=

1− e−(δ+λf )a

a+ 1δ+λf

(1− e−(δ+λf )a), and (17)

ν(a)

v=

δ + λfe−(δ+λf )a

aδ +λf

δ+λf(1− e−(δ+λf )a)

, (18)

where µ is the total mass of employed workers. The employment (vacancy) density is there-

fore increasing and concave (decreasing and convex) in age a. The reason for this is that

for every age a ∈ [0, a) there is a constant number of machines, and older machines have alarger cumulative probability of having been matched in the past. This feature distinguishes

our model from standard-search vintage models where the distribution of vacant jobs is de-

generate at zero and the employment density is decreasing in age a at a rate equal to the

exogenous destruction rate δ.

With the vacancy distribution in hand, we now have the explicit expression for the value

13In Appendix A.2 we describe in detail how to derive (16), (17), and (18).

15

of λw(a),

λw(a; a, λf) = m(θ, 1)δ + λfe

−(δ+λf )a

aδ +λf

δ+λf(1− e−(δ+λf )a)

, (19)

which depends only on the pair of endogenous variables (a, λf), given the relation between

θ and λf .

3.2 Analysis of the equilibrium

We now proceed to show that there exists a unique steady state for the economy with

frictions. We characterize the equilibrium in terms of the rate at which firms find workers,

λf , and the exit age, a. These two variables are jointly determined by the job creation

condition (JC) and the job destruction condition (JD). We begin by studying each of the

two steady-state equations in turn. Next, we turn to the comparative statics of changes

in the unemployment benefits b, the growth rate γ, the interest rate r, and the efficiency

of the matching process (a parameter of the matching function). The formal proofs of our

arguments are contained in the Appendix.

3.2.1 The job creation condition (JC)

The job creation condition states that a potential entrant makes zero profits from setting up

a new machine. We have

Lemma 1. The job creation condition (JC) describes a curve that is negatively sloped in

(a, λf) space.

Lemma 1 follows from the fact that the vacancy value of new firms is increasing in the exit

age a and in the rate at which firms find workers λf . Keeping λf constant, a longer life-span

of capital a increases the vacancy value of a new machine for two reasons: first, it raises the

surplus in every match as explained above, and second, it prolongs the period over which the

new firm can recoup the initial investment. Keeping a constant, a higher rate at which firms

find workers λf also increases the vacancy value of a new machine. The reason is simply

that, almost by definition, a match becomes more likely with a higher λf . Even though

the surplus of a match declines in λf as discussed above, it is straightforward to prove that

this indirect effect is always dominated by the direct effect. The job creation condition thus

defines a curve in (a, λf) space that has a negative slope: if the life-length of a machine goes

16

up, the probability of finding a worker has to go down so that the value of entry remains at

I. This condition is plotted in Figure 1.

Lemma 2. As λf → ∞, the (JC) curve asymptotes to the exit age of the frictionlesseconomy, aCE.

Suppose firms live for a very short period: a is very close to zero. Even if vacant firms meet

workers for sure (with an arbitrarily high rate λf), the life-length of capital is too short for

the initial investment I to pay off. That is, a minimum life-length is necessary to ensure that

the free-entry condition can be satisfied with equality. The asymptote can be worked out to

lie exactly at the destruction age for the competitive solution aCE. Intuitively, as λf →∞,the matching frictions disappear for vacancies and the firms’ entry problem becomes the

competitive problem (2) with solution aCE.

Lemma 3. As a→∞, the (JC) curve asymptotes to a strictly positive value

λminf ≡ (r + δ) rI

(1− β) (1− rI). (20)

Suppose that λf is very close to zero. Even if the life-length of capital is infinite, vacant

firms meet workers with a probability that is too low for the initial investment to pay off in

expected terms. The asymptote value λminf is increasing in I and in the effective discount

rate r + δ, as they both make it more difficult to recover the initial investment, and it is

decreasing in 1−β, the surplus share accruing to the firm. Notice that if rI > 1 (recall that

the condition for existence of the frictionless equilibrium is rI < 1), this asymptote would

be negative.

3.2.2 The job destruction condition (JD)

The job destruction condition states that the productivity of the marginal match at the

cutoff age a equals the flow value of the outside option for the worker.

Lemma 4. If the matching function is Cobb-Douglas, m(v, u) ≡ Avαu1−α, with α > 1/2,

then the job destruction condition (JD) describes a curve that is positively sloped in (a, λf)

space.

The characterization of the job destruction condition (JD) turns out to be a bit more in-

volved. After we multiply the (JD) equation with eγa, we can show that the right-hand side

of the equation is increasing in a and decreasing in λf . Note first that the capital value

17

ln b

CE

JC curve

JD curve

fr I

1 1r I

Figure 1: Job Creation and Job Destruction conditions, plotted in the (λf , a)-space.

of being unemployed depends on the expected surplus from a match, and we know that

the surplus function decreases in λf , as explained earlier. Also, a higher λf decreases the

unconditional meeting probability for workers λw by definition. But there is also a counter-

acting effect that is unique to our model with a vacancy distribution: a faster meeting rate

for vacant firms shifts the vacancy density towards younger vintages with larger potential

surplus. We show that given the assumed Cobb-Douglas matching technology, the value

of search is decreasing in λf because the decline of the unconditional probability becomes

steep enough to overcome the counteracting shift in the vacancy distribution. Intuitively,

one can write λw ' (1/λf)α

1−α so the larger is α, the steeper the decline in λw for a given

rise in λf . For the cut-off age a, a similar argument applies. First, the surplus is increasing

with a. However, the probability of meeting any given vintage–which the surplus function

is weighted by–decreases as a goes up; in particular, it becomes relatively more probable

to meet older vintages, and older vintages have lower surplus than younger ones. The latter

effect is unambiguously dominated by the former effect with the assumed aggregate matching

function. We conclude that the (JD) curve has a positive slope in (a, λf) space (see Figure

1).

18

Lemma 5. As λf →∞, the (JD) curve asymptotes to amax = − ln (b) /γ > 0.

This result tells us that when the meeting frictions disappear, the surplus goes to zero and

output on the marginal job equals the wage, which, in turn, would equal the marginal value

of leisure, given by the welfare benefit b. For the labor market to be viable, we need to

impose the restriction b < 1, where “1” represents the normalized output on the best firm;

otherwise no worker would accept any job.

3.2.3 Existence and uniqueness

Based on our characterization of the (JC) and (JD) curves we can now state a set of conditions

that imply the existence and uniqueness of the steady-state equilibrium.

Proposition 1 An equilibrium with finite values of the pair (a, λf) exists if and only if

rI < 1 and amax > aCE. If the matching function is Cobb-Douglas with α > 1/2, then the

equilibrium is unique.

Proof. We first prove the necessity of each condition. If rI > 1, then no job is created and

the job creation condition is not well defined. As rI → 1, λminf →∞ and the (JC) and (JD)

curves do not intersect for a finite value of λf . If amax ≤ aCE, then the (JC) curve lies strictly

above the (JD) curve, and there is no intersection. Hence, if any of the two conditions of

the Lemma is violated, no equilibrium will exist. To prove sufficiency, it is enough to

consider that if rI < 1, then λminf and aCE are positive and finite, and if amax > aCE, then

the two curves intersect at least once in the positive orthant and an equilibrium¡a∗, λ∗f

¢exists. Furthermore, if the matching function is Cobb-Douglas with α > 1/2, then the (JD)

curve is monotonically increasing, and since the (JC) curve is monotonically decreasing, the

intersection of the two curves (and the equilibrium) is unique.

4 Comparative statics: qualitative results

We now study how technological change and labor market institutions interact in the de-

termination of the equilibrium income distribution and unemployment. In particular, we

are interested in the role of the rate of embodied technological change γ, and the payments

to workers when unemployed b. The parameter b represents the generosity of the welfare

system and simultaneously captures the degree of downward wage rigidity, given the fact

19

that wages in Nash bargaining have the workers’ outside option as a lower bound. We also

study the effects of changes in the interest rate r and in the efficiency of matching A. First,

we analyze the effect of changes in the above mentioned parameters on the equilibrium pair¡a∗, λ∗f

¢, using the job creation and the job destruction curves. We then study the implied

changes for unemployment, wage inequality, and the labor share.

4.1 Comparative statics in (a, λf) space

Lemma 6. A rise in b does not shift the (JC) curve but shifts the (JD) curve downward,

inducing a fall in a∗ and a rise in λ∗f .

The comparative statics of a rise in b are simple: the (JC) curve is unaffected by the worker’s

payoff determinants, and therefore by the unemployment benefit. A higher benefit will

increase workers’ outside options, so in order to restore the (JD) condition, output on the

marginal job has to increase. Hence, for a given value of λf , the exit age a must fall, which

induces a downward shift of the job-destruction curve. Workers become more expensive for

firms without becoming more productive, and therefore machines are scrapped earlier. The

upper panel of Figure 2 shows that the equilibrium moves along the (JC) curve and that both

the life length of firms and labor market tightness thus fall unambiguously. In particular,

the general equilibrium feedback weakens the fall in the life-length of capital, but transfers

part of the impact of the shock on a reduction in firms’ entry.14

Lemma 7. A rise in γ shifts the (JC) curve and the (JD) curve downward, inducing a fall

in a∗. The change in λ∗f is ambiguous.

The comparative statics for γ are somewhat more complicated because an increase in γ

has two counteracting effects on the surplus function (14). First, a higher γ means that a

vintage’s output relative to the frontier falls at a faster rate with age. This obsolescence

effect decreases the surplus of a match. On the other hand, a higher γ reduces the relative

output of the marginal technology of age a and thereby shrinks the outside option value of

a worker. This worker’s outside option effect increases the surplus of a match. The older a

match is the stronger will be the obsolescence effect and the shorter the time period for which

14Upon impact, the higher b leads to a higher wage, lower profits, and shorter job duration; the reductionin firms’ profits, in turn, decreases their incentive to enter the labor market with new machines (λf increases).The implied fall in the meeting rate for workers tends to reduce their outside option and hence their hiringcosts and increase profits, therefore making a smaller fall in a necessary for the adjustment to the newequilibrium.

20

it will benefit from the worker’s outside option effect. We show that there is a critical age

such that for vintages younger (older) than this critical age the surplus rises (falls) with γ. In

particular, at age zero there is no obsolescence effect, so the surplus of a new machine grows

unambiguously with γ; this last remark will be important later, once we compare our model

with the standard Aghion-Howitt/Mortensen-Pissarides framework where all vacancies are

of age zero.

Notwithstanding this non-monotonicity of the surplus function, we can prove that the

shifts of the (JC) and (JD) curves are unambiguous. A rise in γ increases the value of

new vacancies and thus shifts the (JC) curve downward: for a given scrapping age, a lower

meeting rate for vacant firms is necessary to bring the value of vacancies back in line with

the constant set-up cost I. If we turn to the (JD) curve, a rise in γ reduces output on the

marginal job, but also increases the value of search for an unemployed worker, as waiting is

compensated by the expectation of being matched to a more productive firm. Both effects

lead to the conclusion that for the (JD) condition to hold for a given market tightness,

the marginal machine has to be scrapped earlier so the curve will shift downward. Taking

these two shifts together, we see that the life length of firms declines unambiguously with

a higher rate of technological change. Whether labor market tightness goes up or down

depends on the relative slopes of the two curves and the initial position of the curves, that

is, on the “starting values” for the parameters, including the initial growth rate (see the

lower and central panels of Figure 2). This is an important factor for understanding the

complementarity of growth and institutions, as explained below.

Lemma 8. A rise in r shifts the (JC) and (JD) curves upward, inducing an increase in a∗.

The effect on λ∗f is ambiguous.

An increase in the interest rate lowers the weight on future profits and thus lowers surplus.

This surplus reduction makes the job destruction curve shift up: for a given λf , the worker

who is indifferent between staying on the job and leaving now needs a longer life on the

job to counteract the fall in the surplus of the job. Similarly, the job creation curve shifts

up: a lower surplus must be counteracted by a longer life in order for the firm to remain

indifferent between entering and not entering. As a result, higher interest rates induce a rise

in a, whereas the impact on λf cannot be signed.

21

•

•

f

E

JD

EJC

Curve shifts after a rise in b

•

•

f

E

JD

JC

Curve shifts after a rise in γ:Response along the incidence marginE

•

•

f

E

E

JD

JC Curve shifts after a rise in γ:Response along the duration margin

Figure 2: Qualitative comparative statics with respect to b (top graph) and γ (middle andbottom graph). The label E refers to the initial (pre-shock) equilibrium and the label E0 tothe final equilibrium. The dotted lines represent the JC and the JD curves after changes inb and γ.

22

The severity of the matching friction can be regulated with the level of the shift parameter

of the matching function (A in the Cobb-Douglas formulation).

Lemma 9. In the limit, as A → ∞, the frictions disappear, and the equilibrium with

frictions converges to the competitive equilibrium.

As A → ∞, the matching friction vanishes and the equilibrium of the economy entails

a→ aCE and λf →∞. Recall that, when λf →∞ in the standard matching model without

vacancy heterogeneity, a → 0: the vintage structure vanishes without frictions. In our

model, the vintage capital structure survives in limiting frictionless equilibrium. Frictions

extend the life of capital, but are not necessary for old machines to be operated by workers

in equilibrium.

4.2 Unemployment, inequality, and labor share as functions of(a, λf)

Having characterized the changes in the equilibrium pair¡a∗, λ∗f

¢for a given parameter

change, we can study how the change in¡a∗, λ∗f

¢together with the underlying parameter

change determines the labor market outcomes in which we are interested, namely unemploy-

ment, wage inequality, and the wage-income share.

In steady state, the flow into unemployment equals the flow out of unemployment. That

is,

δµ+ µ(a) = λwu = m(θ, 1)u. (21)

To understand how unemployment responds to changes in the pair (a, λf), it is convenient

to restate (21) asu

1− u=

δ + µ(a)/µ

m(θ, 1), (22)

which is simply the product of unemployment incidence and duration. The degree of en-

dogenous job destruction µ(a)/µ, that is, the fraction of matched jobs destroyed at a, can be

read in (17). A rise in a reduces the unemployment rate, since endogenous job destruction is

reduced. A rise in λf has two counteracting effects. First, a higher λf reduces the meeting

probability for unemployed workers, which in turn increases unemployment duration. Sec-

ond, a higher λf reduces endogenous job destruction, which in turn reduces unemployment

incidence. As λf increases, vacant firms meet workers at a faster rate, so the employment

distribution shifts towards younger machines, and there are relatively fewer machines at the

23

exit age. We can show that for the Cobb-Douglas matching function with α > 1/2, the first

effect dominates and unemployment increases with λf .15

Wage payments support the surplus-sharing allocation in the economy with frictions.

Embodied technical change therefore generates wage inequality since it implies productiv-

ity differences across vintages. Using the surplus-based definition (10) of the value of an

employed worker W (a) in equation (8) and rearranging terms, we obtain the wage rate as

w (a) = (r − γ)U + β [(r − γ + δ)S (a)− S0 (a)] .

Using the differential equation for the surplus (11), we obtain the wage equation

w (a) = (r − γ)U + β£e−γa − (r − γ)U − λf (1− β)S (a)

¤. (23)

The Nash wage rate exceeds the flow value of unemployment by a fraction β of the quasi-

rents. This latter term is composed by the production flow e−γa, net of the worker’s flow

outside option (r−γ)U and net of the firm’s expected surplus share of being in an alternativematch λf (1− β)S (a). The last term is age-specific and it is intrinsically related to the

value to older firms of becoming vacant. In standard models, this value is zero for every

firm. The wage equation also confirms that at the separation age a, the firm and the worker

are indifferent between continuing the match and separating. Evaluating (23) at a together

with the destruction condition shows that w (a) = e−γa, that is, the flow profits are zero. It

also demonstrates that w (a) = (r − γ)U , that is, the worker is indifferent between working

and entering unemployment.

Wage inequality in the economy is determined by two factors: the maximum wage dif-

ferential w(0)/w(a) and the employment distribution. We can write the maximum wage

differential asw(0)

w(a)= (1− β) + β

1− λf(1− β)S(0; a, λf)

e−γa. (24)

A longer life-span for the job increases the distance between the highest and lowest produc-

tivity in the economy, thus raising wage inequality: the technological heterogeneity effect.

We explain in the proof of Lemma 9 that as the meeting rate for firms λf → ∞, the flowoutside option for the new firm grows towards the highest possible flow profit 1− e−γa, thus

the term multiplied by β in (24) converges to one, implying (as in the competitive economy)

15To show this result, multiply both numerator and denominator of (22) by λf and differentiate u/(1−u)with respect to λf .

24

perfect wage equality: the firm’s outside option effect. The intuition is that as λf increases,

firms meet at a faster rate and their bargaining position improves so much that gradually

workers are squeezed against their outside option, which is constant, so wage inequality falls.

The changes in the employment distribution are also crucial for equilibrium inequality

because, given that the wage rate (23) has a constant component and a component linked to

the productivity of the vintage, younger vintages display larger inequality. As shown before,

an increase in λf or a reduction in a shifts the employment distribution towards younger

vintages, with more inequality: the distributional effects.

Finally, consider the labor share.16 A shorter life-length of capital a unambiguously

increases the wage share through a rise in the equilibrium outside option of the worker e−γa.

Instead, a larger firm’s meeting rate λf improves the firm’s threat point in the bargaining

and reduces the wage share of output in each match. The distributional effects following

an increase in λf or a reduction in a shift the employment distribution towards younger

vintages, which have smaller labor share of output (recall that on matches of age a the labor

share is one).

4.3 How inequalities are affected by growth and institutions

What are the qualitative effects on labor market variables of changes in the welfare system

and the rate of technological change? For our model we find that a more generous welfare

system increases unemployment, tends to reduce wage inequality, and is likely to have no

impact on the labor share. We also find that a faster rate of technological change tends to

increase unemployment and wage inequality and tends to reduce the labor share.

A more generous welfare system, higher b, decreases a∗ and increases λ∗f , which leads to a

rise in the unemployment rate: both incidence and duration increase (see the upper panel of

Figure 2). Economies with higher b should display less wage inequality. Because technological

heterogeneity declines (a falls) and the firm’s outside option increases (λf increases), the

maximum wage differential across vintages falls, which tends to reduce inequality across

vintages. There is a countervailing effect since the new equilibrium employment distribution

gives more weight to younger vintages, which display more wage inequality. Barring strong

changes in the employment distribution, however, inequality will fall with b. Welfare benefits

16A closed-form expression for the labor share in our model can be obtained, but it does not add much tothe intuition we have built in the previous analysis on the wage rate and on inequality, so we omit it.

25

have conflicting effects on the labor share, as explained above, so we should not expect large

differences across economies with different b.

A faster rate of embodied technological change, higher γ, lowers a∗ and has an ambiguous

effect on λ∗f . The shorter lifetime of firms increases the unemployment rate through a higher

unemployment incidence. The ambiguity with respect to the worker-finding rate can be

understood by looking at the two extremes depicted in Figure 2. We could be in an economy

that responds to the shock with a sharp fall in a but no significant change in λf , generating

a higher unemployment incidence, but little change in unemployment duration (the central

panel of Figure 2). Alternatively, we could be in an economy where job separation rates (a)

are barely affected and all the adjustment takes place through a lower entry rate of firms

(λf), that is, unemployment duration rises with little impact on unemployment incidence

(the lower panel of Figure 2).

A faster rate of embodied technological change directly increases wage inequality. This

is counteracted by the implied decline of the exit age a. Furthermore, if the firm’s contact

rate λf rises strongly, it will create an additional tendency towards lower inequality through

the firm’s outside option effect. Overall, we expect the direct effect to dominate, but wage

inequality will tend to increase more in economies that respond through the unemployment

incidence margin (λf) rather than the unemployment duration margin (a).

A faster rate of embodied technological change tends to reduce the labor income share.

The direct effect of a rise in γ reduces the equilibrium value of unemployment, e−γa, and the

share of output going to labor in each job. The indirect effect through a and λf depends

crucially on the margin of adjustment. In economies that adjust through the unemployment

incidence margin, the substantial shortening of job durations tends to counteract the direct

effect and increases the labor share. In economies that adjust through the unemployment

duration margin, the substantial increase of the firm’s contact rate improves the firm’s outside

option value and reduces the worker’s share in production. Finally, as explained, changes in

the employment distribution always reinforce the fall in the labor share. Thus, we should

expect a more dramatic fall in the labor share in economies responding through the duration

margin.

Finally, consider the interaction of labor market institutions and technological change.

We argue that in our model labor market institutions can, at least qualitatively, account

for a differential response in labor market variables to the same acceleration in embodied

26

technological change. Consider first a low-benefits economy (the United States). An accel-

eration in the rate of productivity growth of new vintages represents an obsolescence shock

that makes installed capital obsolete faster —labor costs grow at a swifter pace over the life

of a job with fixed productivity. Firms respond to the obsolescence shock by adopting new

technologies more rapidly, and in order to do that firms must shorten the optimal life-length

of machines. The U.S. economy reduces the lifetime of machines and adjusts along the

unemployment incidence margin discussed above (see the central panel in Figure 2). Now

consider the response of a high-benefits economy (Europe). In our previous discussion we

argued that higher benefits move the initial equilibrium down the job creation curve towards

its flat region (see the bottom panel in Figure 2). With an initial position where the job

creation curve is very flat, a rise in the growth rate γ will induce a much larger rise in λf .

The logic for this result is straightforward. High benefits and high labor costs have already

pushed the optimal life-length of capital very close to its technological minimum aCE, and

the life-time of a machine cannot be reduced much further. Since operating firms cannot

decrease a any more, they need to be compensated in a different way, i.e. through an increase

in their contact rate when positions are vacant. The corresponding stronger decrease in the

worker’s meeting rates induces a larger rise in unemployment duration, a smaller increase in

wage inequality, and a larger decline in the labor share of aggregate income. To conclude,

the level of the policy determines the location of the pre-shock equilibrium and this, in turn,

determines the nature of the adjustment.

5 The quantitative role of technology-policy comple-

mentarity

We now go beyond a purely qualitative analysis. We will organize the analysis around the

United States—Europe comparison; a by-product of this calibration analysis will be the an-

swer to our first quantitative question: what is the role of the mechanism we study for

residual wage inequality and unemployment? The United States—Europe question is: can

our simple model account quantitatively for the differential behavior of unemployment, wage

inequality, and income shares in the United States and Europe over the past thirty years?

In our experiment we calculate the steady-state responses of the model economies to the ob-

served increase of the rate of embodied technological change γ. The model economies differ

27

with respect to the policy measure b, which we interpret as a form of welfare benefit and/or

downward wage rigidity. We find that the same increase of the rate of technological change

implies a larger increase of the unemployment rate and a smaller increase of wage inequal-

ity in economies with high welfare benefits. Quantitatively, the differential unemployment

response of high and low welfare payment economies calibrated to continental Europe and

the United States is remarkably similar in magnitude to the actual differential response of

these economies.

The benchmark model does not match the differential response of the labor income share:

although it predicts that the labor income share will decline, the magnitude of the decline is

the same for all economies. We consider the potential of another difference in labor market

institutions–employment protection–to account for the differential response in the shares

of labor income. Employment protection legislation tends to be much stricter in continental

Europe than in the United States. In Section 5.3, we study the effect of a simple version

of employment protection and we find that stricter employment protection in continental

Europe can account for the relatively larger decline of the labor income share in Europe.

5.1 Calibration

The quantitative analysis requires calibration of the model economy. In the calibration,

we choose to match U.S. averages for the pre-1970 period since the technological shock we

model is likely to have hit the economy around the early-mid 1970s.17 Moreover, initially we

choose to represent Europe as an economy that differs from the United States only in terms

of policy. This choice simplifies the interpretation of the results since the different outcomes

are entirely attributable to different policies. Later, we relax this assumption. Finally, it

is important to point out that in the experiment we treat the data for the late 1960s and

mid-1990s as both representing steady states.

Given the choice of a Cobb-Douglas matching function with constant elasticity of match-

ing with respect to unemployment equal to α and scale parameter A (that we normalize

to 1), the model has seven parameters: {r, δ, α, β, I, b, γL} . We set r to match an annualinterest rate of 7%.18 We set δ in order to match an annual worker’s separation rate from

17See Hornstein and Krusell (1998) and Acemoglu (2000) for discussions of the timing of the technologicalacceleration.18This value of r is slightly larger than the one commonly used in the literature, but it is necessary to

keep r > γ in every experiment so that the infinite discounted sums are all well defined. In Section 5.4 we

28

employment to unemployment equal to 25%, as reported in CEPR (1995, page 10).19 We

choose α to match an average unemployment duration of approximately 8-9 weeks (as re-

ported by Abrahams and Shimer [2001]), which together with the above separation rates

gives us an unemployment rate of 4%, the U.S. value for the early 1970s (as reported in

Table 1). The Nash bargaining parameter β is chosen to match a labor share of 0.69, and

the cost of setting up a production unit I is chosen to reproduce an average age of capital of

about 11.4 years, as reported by the Bureau of Economic Analysis (1994) for the late 1960s.

The speed of embodied technological change γ is matched to the inverse of the rate of

decline of quality-adjusted relative price of equipment before and after the mid-1970s. This

procedure implies a value of 3.5% per year for the first steady state with a low rate of embod-

ied technical change (γL). As documented in Gordon’s (1990) influential work on quality-

adjusted prices for durable goods, and more recently by Cummins and Violante (2002), in

the last two decades the speed of embodied technical change has increased substantially to

reach 6.5% in the years 1995-2000.20 How reasonable is it to assume that the shock is com-

mon between the U.S. and Europe? A recent OECD study (Colecchia and Schreyer 2001)

measures the decline in relative price for several high-tech equipment items across various

countries in Europe from 1980 to 2000. Table 2 shows that in the last decade large European

countries experienced an acceleration quantitatively comparable to the United States. Since

high-tech goods drove the technological acceleration in the aggregate price index, we can be

confident that the aggregate indexes should display similar patterns.21

discuss the impact of changes in r.19In the model, the separation rate, i.e., the unconditional probability that a worker separates from a job

within the period, is defined as [δ + µ (a)] /µ. Note that it would be incorrect to match this variable tojob destruction rates (i.e., job flows rather than worker flows, as we do) since the event occurring at rate δinvolves only a separation of workers and machines, but not the destruction of the job.20Other authors, using measurement techniques different from quality-adjusted relative prices, arrived at

very similar conclusions on the pace of embodied technical change in the postwar era (see for example Hobjin2000) for the United States.21Ideally, one would like to compare growth rates in the 1970s as well, but these are not available for

European countries. Table 1 shows that in some continental European countries the measured accelerationis even larger than in the United States, but one should keep in mind that the high-tech goods’ share ofaggregate equipment in these same countries is likely to be smaller than in the United States.

29

Table 2: Acceleration of capital-embodied technical change

U.S. UK France Germany Italy

Computers 7.6 7.3 7.4 4.8 8.7

Communications 4.4 5.1 5.5 2.6 7.4

Software 1.9 2.8 3.0 2.2 5.1

Note: Difference between the rate of decline of quality-adjusted

relative price of equipment-capital type in the period 1990-2000

and the period 1980-1990.

Source: Table 4, Colecchia and Schreyer (OECD 2001)

In our experiment we gradually increase the annual growth rate γ from 3.5% to 6.5% and

study the response of unemployment, wage inequality, and the labor share for economies with

different values of the benefits parameter b. This simple parameter is supposed to summarize

a wide variation of benefit policies with respect to unemployment duration, family situation

(none of which we model), and country. The OECD Employment Outlook (1996) computes

average replacement rates from unemployment benefits in OECD countries from 1961 to

1995 for two earnings levels, three family types, and three durations of unemployment. In

the mid-1970s the OECD average replacement rates for the United States were 11%, whereas

for many European countries the replacement rates were 40% or higher in the same time

period (Chart 2.2, page 29).22

The OECD replacement rates for Europe understate the measure of benefits that we

use in our model, for many European countries offer long-term social assistance schemes in

addition to unemployment benefits. Our parameter should reflect these policies since most

of them are not earnings-related and have indefinite duration. Hansen (1998) computes

corrected replacement ratios to account for social assistance and finds much larger values for

a set of European countries, all between 45% and 72% (Hansen 1998, Graph 3, page 29).23

In the baseline economy, which we interpret as the U.S. economy before the technological

acceleration, we set b = 0.05, which implies a ratio of welfare benefits to average wage of

roughly 10%. To model European-type economies, we gradually increase b to 0.4, which

implies a ratio of welfare benefits to average wage of roughly 70%. Finally, although in

the data there is some time-series variation in these replacement rates, we model them as

constant through time, that is, the only source of shock is the rate of productivity growth

of capital. Bertola, Blau, and Kahn (2001) and Blanchard and Wolfers (2000) find that

22The same OECD data source documents that average replacement rates reached peaks of 50% in theNetherlands, 45% in Belgium, 37% in France, 35% in Spain, and 30% in Germany.23These replacement rates are calculated for a 40-year old single male production worker.

30

time-variation in labor market institutions is small compared to cross-country differences

and empirically accounts for a minor fraction of unemployment rate differentials.

The calibrated parameter values are summarized in the following table:

Table 3: Calibration of the Model Economy

Parameter Value Moment to match

r 0.017 interest rate

δ 0.0515 separation rate (CEPR 1995)

β 0.50 labor share (Cooley 1995)

α 0.55 unemployment duration (Abrahams and Shimer 2001)

I 14.5 average life of capital (BEA 2001)

b 0.05-0.4 welfare benefits (OECD 1996 )

γL 0.009-0.016 relative price of equipment (Krusell et al. 2000)

Note: A unit time period represents one quarter.

Finally, we should stress that we have not used any of the parameters to try to match the

initial level of wage inequality in the data. The reason is that wage inequality in this model

is purely due to vintage capital effects and we are not aware of data counterparts measuring

the extent of inequality that can be attributed to this source. One contribution of our work

is that we can use the calibrated model as a measurement tool to find out how much wage

inequality is generated by this mechanism in a United States-like economy. We return to

this point in the next section.

5.2 Results

The main quantitative results of our experiment are reported in Figure 3, where a number of

equilibrium outcomes of the model (unemployment rate, unemployment duration, separation

rate, maximum age of capital, wage inequality, and labor share) are plotted for different

rates of embodied technical change (from 3.5% to 6.5%) and for different levels of the policy

variable of interest b (from 0.05 to 0.40).

The contribution of vintage capital and matching frictions to wage inequality.

Let us start with wage inequality, measured by the 90-10 log-wage differential. As ex-

plained, the model is designed to generate inequality among ex ante equal workers, which

originates from a combination of labor market frictions and vintage capital differentials.

Figure 3 shows that in the final steady state of the baseline economy the 90-10 log-wage

31

differential is around 7%. An alternative way to measure inequality induced by this mecha-

nism is to say that a vintage differential of 10 years in the capital used by firms translates

into a wage gap of about 6% in our model economy. How reasonable is this number? We

are not aware of any exact data counterpart, but Doms, Dunne, and Troske (1998) provide

an interesting benchmark of comparison. They examine a sample of U.S. plants in 1988 for

which one can observe both the degree of technological advancement of the plant (i.e., the

technologies recently adopted) and the characteristics of the workers (like education) and

conclude that the wage differential induced only by the technological gap between a plant

in the top quartile and a plant in the bottom quartile of the technological scale is 8.4% for

production workers and 12.7% for technical and non-production workers (Table III, page

267). These figures seem to suggest that our estimates are of the right size.

How much do vintage capital and matching frictions contribute to overall wage inequality

among ex ante identical workers in the U.S. economy? According to Katz and Autor (1999)

the 90-10 log-wage differential of “residual” wage inequality, that is inequality not related to

observable characteristics, such as age and education, is about 90%. Gottschalk and Moffitt

(1994) argue that the fraction of residual inequality accounted for by permanent unobservable

characteristics of individuals, that is “innate ability”, is roughly 2/3. Thus, the model tells

us that in the United States labor market frictions, together with vintage capital, account

for almost 25% of wage inequality among ex-ante equal workers. The remaining inequality

can be the result of match quality and skill dynamics within and between jobs.

The impact of a faster rate of technological change on labor market inequalities.

The effect of a faster rate of technological change on wage inequality is ambiguous and

depends on the magnitude of the policy variable b. Figure 3 shows that wage inequality in

the U.S.-type economy rises by 1 point. As explained earlier, this increase in inequality is

brought about essentially by an increase in technological heterogeneity through the interplay

between a higher γ (a large productivity differential across successive vintages of machines)

and a lower a (a lower age gap between the youngest and the oldest machine; see Figure

5). In the simulations, the first effect dominates, and the distributional effects are always

fairly small. The firm’s outside option effect restrains inequality from a sharp surge. In

economies with a large b, the increase in λf is massive, so this latter effect is very strong and

in some extreme cases (e.g., b = 0.4), can dominate the larger technological heterogeneity

32

3.5 4 4.5 5 5.5 6 6.53

5

7

9

11

13

15

17

19

21

23

25

27

γ

UNEMPLOYMENT RATE

b=.05b=.15b=.30b=.40

3.5 4 4.5 5 5.5 6 6.58

13

18

23

28

33

38

43

48

53

58

63

68

73

γ

UNEMPLOYMENT DURATION (weeks)

3.5 4 4.5 5 5.5 6 6.524.5

25

25.5

26

26.5

27

27.5

γ

ANNUAL SEPARATION RATE

3.5 4 4.5 5 5.5 6 6.57

7.5

8

8.5

9

9.5

10

10.5

11

11.5

12

γ

AVERAGE AGE OF CAPITAL (years)

3.5 4 4.5 5 5.5 6 6.50.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

γ

90-10 LOG WAGE DIFFERENTIAL

3.5 4 4.5 5 5.5 6 6.50.6

0.615

0.63

0.645

0.66

0.675

0.69

γ

LABOR SHARE

Figure 3: The effects of a rise in the rate of embodied technical change γ in the vintage-capitalmodel when the economies differ only via the level of welfare benefits b.

33

and lead wage inequality to fall.24 The model suggests that wage inequality did not increase

in continental Europe as a response to the shock because the sharp increase in unemployment

duration improved the bargaining position of the firms so as to squeeze workers against their

outside option, which is invariant. It is worth remarking that this argument, based on the

firm’s outside option, is a unique feature of our model with vacancy heterogeneity.

A faster rate of technological change increases unemployment for all values of the welfare

benefits, but the increase is much more pronounced for European-type economies with high

b. If we take the cautious view that the new steady-state level of capital-embodied produc-

tivity growth is 5.5%, then the model predicts that in the baseline economy unemployment

moves very little, by less than 1 point, whereas for b = 0.40 unemployment jumps by 6

percentage points. It is clear from Figure 3 that although both unemployment duration

and the separation rate rise with γ, the bulk of the differential increase in unemployment is

explained by the former: the separation rate changes only very marginally, by roughly 1%

in all the economies considered, while the increase in unemployment duration is small in the

U.S.-type economy (from 8.5 to 10.5 weeks) but is substantial in the economies with high

b, from 19 to 38 weeks when b = 0.4. This result is consistent with the recent experience

of European labor markets, where labor turnover did not increase significantly, and where

most of the increase in the unemployment rate is associated to longer durations (Machin

and Manning 1999). Overall, quantitatively, the differential rise in unemployment rate is of

an order of magnitude comparable to the data in Table 1.

The labor income share (last panel of Figure 3) declines as γ increases, independently of

the magnitude of the policy parameter b. Note also that the model generates labor shares of

a similar size independently of the magnitude of the policy parameter b, whereas in Europe

the labor share has been slightly above the US value until 1980. This independence reflects

the offsetting effects of b on the labor share discussed in Section 4.3. As expected, the labor

share falls with γ: a rise from 3.5% to 5.5% reduces the labor share in every economy by

6%. The magnitude of this decline of the labor share is in line with the data for continental

European economies in Table 1 (6 points). On the other hand, the size of the US labor

income share decline in the data is substantially smaller than is predicted by the model.

In the next section, we argue that the inclusion of another important labor market policy,

24Note that this decline in wage inequality is not inconsistent with the European data. Table 1 shows thatwage inequality fell on average in Europe from 1980 to 1990, and in some countries such as Germany andNorway kept falling until the mid 1990s.

34

employment protection legislation, helps the model match the differential decline in labor

shares.

One limitation of our experiment is that the model economies initially (i.e., for a low

value of γ) display a large unemployment differential since they only differ through the

policy b. The data in Table 1 show that the unemployment rates in continental Europe and

the United States were quite similar in the early 1970s. Machin and Manning (1999) report

that although they had a similar unemployment rate in that period, unemployment duration

was already much longer in continental Europe (Machin and Manning 1999, Table 4, page

3100). To account for this observation, we modify the experiment and change the separation

rate δ with the policy parameter b to keep the initial unemployment rate constant at 4%.

The results are in Figure 4: the rise in unemployment is still magnified by the policy b by an

amount which is in line with the data, and the bulk of the rise is once again due to longer

durations, with the separation rates changing very little. The changes in wage inequality

and labor share remain of the same magnitude.

The model’s implications for labor market inequalities are closely related to its predictions

for the economic lifespan of capital. In the wake of a technological acceleration the age of

capital declines in the model: firms scrap their machines earlier in response to a faster

obsolescence rate.The Bureau of Economic Analysis publishes data on the age of capital for

the U.S. economy since 1925. In Figure 5 we plot the average age of private fixed assets

(BEA 2002, Table 2.10) for our sample period 1965-1995. Average age in the United States

falls from 11.4 years in 1965 to 8.6 in 1985 and then it rises again to 9.5 years in 1995. The

model’s age of capital (Figures 3 and 4) falls to 8.7 as γ approaches 5.5%. Overall, the size

of the age decline in the model is quite similar to that in the United States, with the U.S.

data showing a fall by 17% and the model by 25%.

5.3 Extension: employment protection legislation

Our analysis of differences in labor market institutions has focused so far on the role of

welfare benefits/unemployment insurance. This analysis has successfully accounted for the

differential response on unemployment and wage inequality in Europe and the United States

to an increase of the rate of technological change. Differences in welfare benefits, however,

cannot account for the differential response of the labor income share in these countries. We

now assess whether differences in employment protection legislation, in particular a firing

35

3.5 4 4.5 5 5.5 6 6.53

4

5

6

7

8

9

10

11

12

13

14

15

γ

UNEMPLOYMENT RATE

b=.05b=.15b=.30b=.40

3.5 4 4.5 5 5.5 6 6.58

13

18

23

28

33

38

43

48

53

58

63

68

73

γ

UNEMPLOYMENT DURATION (weeks)

3.5 4 4.5 5 5.5 6 6.59

11

13

15

17

19

21

23

25

27

γ

ANNUAL SEPARATION RATE

3.5 4 4.5 5 5.5 6 6.57

7.5

8

8.5

9

9.5

10

10.5

11

11.5

12

γ

AVERAGE AGE OF CAPITAL (years)

3.5 4 4.5 5 5.5 6 6.50

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

γ

90-10 LOG WAGE DIFFERENTIAL

3.5 4 4.5 5 5.5 6 6.50.6

0.615

0.63

0.645

0.66

0.675

0.69

γ

LABOR SHARE

Figure 4: The effects of a rise in the rate of embodied technical change γ in the vintage-capital model when the economies differ both via the level of welfare benefits b and theexogenous separation rate δ.

36

1965 1968 1971 1974 1977 1980 1983 1986 1989 1992 19958

8.25

8.5

8.75

9

9.25

9.5

9.75

10

10.25

10.5

10.75

11

11.25

11.5

11.75

12

Year

Figure 5: Average age of capital in the U.S. economy (in years), 1965-1995. Source: Bureauof Economic Analysis (2002), Table 2.10.

tax, can account for the differential response of the labor income share.

Introducing a pure firing tax complicates the analysis significantly. The destruction age

for a vacancy, a, would no longer be the same as the destruction age for an existing match,

a. This is because prior to matching, the cost of the dissolution of the match in the future–

the firing tax–is not a liability, and it will not become one until the match is formed. An

existing match treats this cost as sunk and the match dissolves at a capital age such that the

total surplus of the match (which includes a liability T to the government) equals −T : atthis point the marginal profit flow from production is equal to zero. A vacancy would not be

posted at such an age because the total surplus is negative, so vacant capital withdraws from

the market at an earlier age than matched capital: a < a. Formally, the presence of a firing

tax leads to two different notions of surplus: surplus upon hiring, where the disagreement in

the bargaining does not imply the payment of the tax, and surplus during the match, where

it does. As a result, we have a two-tier labor market (with two wage functions) and two

destruction thresholds. The structure of the model with a simple firing tax is therefore quite

different from the one studied above.

37

Fortunately, there is a simple way to introduce an employment protection policy without

affecting the structure of our equilibrium. Consider an employment protection policy that

combines a firing tax with a hiring subsidy. In particular, assume that in an existing match

the firm pays a firing tax T on separation, and a vacant firm that hires a worker receives

a hiring subsidy T .25 In Appendix A.6 we solve the model and show that the modified

destruction rule for a match is

e−γa + (r − γ)T = (r − γ)U. (25)

The intuition behind this equation is easy to grasp once it is understood that the policy T

has the form of a zero-coupon government bond from which the firm is entitled to receive the

growth-adjusted return, r − γ, for the duration of the match. In other words, every period

the firm’s payoff from the match is augmented by the amount (r − γ)T , which will tend to

extend the life-length of capital.

In the Appendix we show that the new wage function is

w(a) = (r − γ)U + β£e−γa + λf(1− β)S(a)− (r − γ)(U − T )

¤. (26)

As before, we have w(a) = (r − γ)U , and a separation will occur when the marginal oper-

ating profit of the firm e−γa + (r − γ)T is entirely paid into the wage bill. The job creation

and job destruction equations become

e−γa + (r − γ)T = b+ β

Z a

0

λw(a; a, λf)S(a; a, λf)da, (JD’)

I = λf(1− β)

Z a

0

e−(r−γ)aS(a; a, λf)da. (JC’)

One can see that the comparative statics of the job creation curve (JC’) are unchanged. The

comparative statics of the job destruction curve (JD’) with respect to γ are also qualitatively

unchanged. Now, however, a rise in γ reduces the LHS of (JD’) by an amount that increases

with the size of T , so the policy amplifies the downward shift of the (JD’) curve. Here the

policy-technology complementarity is very stark. Equation (26) makes clear that the larger

the tax/subsidy T , the more the wage will fall as γ increases: through bargaining, the worker

can appropriate a share β of this additional return to the match, and the fall in this quasi-rent

due to an increase in the growth rate γ is proportional to T . This mechanism will tend to

25This tax/subsidy scheme implies that every period the policy satisfies a balanced budget constraint forthe government because total separations equal newly created matches in steady-state.

38

reduce the labor share more severely in economies with more generous employment protection

policies. Finally, as we should expect, a rise in the firing tax (for a given γ) shifts the (JD’)

curve upward, inducing longer job tenures and more firm entry, which unambiguously lowers

unemployment.

We now move to the quantitative analysis. Based on the data on firing costs reported

in the OECD Employment Outlook (1999), we choose a conservative range for the tax

T running from zero to one year of salary.26 The results are displayed in Figure 6. In

economies with a high firing tax T , the labor share starts at a higher level and falls much

faster as γ increases, following the same pattern as in the data: the firing tax can account

for a 4.5 percentage point differential decline across economies, which is very close to the

number implied by Table 1, 4.3%. Although the model economy is able to generate this

differential fall, in absolute terms it overpredicts the decline in the labor share. For the

United States, it predicts a fall of 6%, whereas the decline in the labor share in the U.S. data

is only 1.5%. Interestingly, the firing tax is much less important than the welfare benefits

in the determination of cross-country differences in the evolution of the unemployment rate:

independently of the level of T , unemployment duration and separation rates change by very

similar amounts in the model.

In conclusion, employment protection does not seem to be responsible for different pat-

terns of unemployment between U.S. and Europe, but it might be important in understanding

the different evolutions of the distribution of income between capital and labor.

5.4 Discussion of related results in the literature

Ljungqvist and Sargent (1998) are among the first to study quantitatively the channel of

technology-policy complementarity to explain the relative labor market performances of the

U.S. and Europe. They model the common shock as a rise in the degree of skill depreciation

of unemployed workers and analyze the impact of this shock in a search model where workers

receive unemployment benefits linked to their past earnings (and their past skills) and receive

wage offers linked to their current skills. More rapid skill depreciation worsens the value of

the average wage offer compared to the value of unemployment and increases unemployment

duration. This model has a fixed wage distribution and fixed number of jobs; thus, the

26In virtually all OECD economies, firing costs are proportional to the wage at separation, so we modelT = (r − γ)U · t. We have verified that this choice has no impact on the results. The other parameters ofthe model are unchanged with respect to the benchmark calibration. In particular, we set b = 0.05.

39

3.5 4 4.5 5 5.5 6 6.50.6

0.61

0.62

0.63

0.64

0.65

0.66

0.67

0.68

0.69

0.7

0.71

0.72

0.73

0.74

γ

t=0t=2t=4

Figure 6: The response of the labor income share to a rise in the rate of embodied technicalchange γ in the vintage-capital model when economies differ with respect to the firing tax t.

mechanism operates entirely on the labor supply side. In our model, in contrast, workers

accept every job offer, but both wages and labor demand (i.e., the number of jobs) are

endogenous. In this sense, the two papers highlight the importance of the complementarity

between technological shocks and welfare benefits along two parallel margins: labor supply

and labor demand.

Blanchard and Wolfers (1999) attribute unemployment differentials across countries and

over time in a panel of OECD countries to the interaction between shocks and labor market

policies. They consider three types of shocks–a productivity growth slowdown, a rise in

the interest rate, and technological change biased against labor–together with several types

of policies, including unemployment insurance and employment protection legislation. Our

embodied productivity acceleration can be interpreted as the source of technological change

biased against labor, measured by Blanchard and Wolfers directly off the fall in the labor

share. The authors find significant evidence of interactions between shocks and institutions:

in line with our findings, they report that a shock that increases unemployment by 1% in the

country with the lowest welfare benefits would have an impact 5 times larger in the country

40

with the most generous welfare payments, whereas this “multiplier” effect for employment

protection legislation is only 2.27 Bentolila and Saint-Paul (1999) also study the evolution

of the labor share across OECD countries since 1970. They find that in the presence of

institutions that promote wage rigidity, shocks that reduce employment significantly also

reduce the labor share of income. One can view our quantitative study as the “structural”

counterpart of these empirical analyses.

In a recent paper, den Haan, Haefke, and Ramey (2001) study the quantitative implica-

tions of interest-rate and TFP shocks within a calibrated version of the traditional Mortensen

and Pissarides (1994) framework. In this class of models, a rise in the real interest rate or

a fall in TFP have identical effects: the equilibrium unemployment rate increases through

a rise in the “effective discount factor,” as demonstrated already in Pissarides (1990). Like

us, den Haan et al. also ask if institutions can account for the differential response of labor

markets to these shocks. They find that labor market institutions are important only if the

United States and Europe differ substantially in their cross-sectional distributions of match-

specific productivities, a dimension of the data that the authors do not attempt to calibrate.

In our model economies, the effects of an interest rate shock are negligible. Following an

interest rate hike, fewer jobs are created and unemployment duration rises, but at the same

time the destruction age increases, which reduces the separation rate (see Lemma 8). We

find that the net effect on unemployment is small and the policy-shock interactions are much

less pronounced than in our experiments. For example, as the real interest rate grows from

1% per year to 6%, the differential rise in the unemployment rate between the economy with

the highest benefits and the economy with the lowest benefits is only by 2%.28

Our paper relates to the argument advanced for example in Blanchard (1997) and Ca-

ballero and Hammour (1998) whereby European unemployment is largely due to expensive

labor services and a fall in labor demand associated to firms’ adoption of ever more labor-

saving technologies. Our model does not allow for any substitutability between capital and

labor at the level of the production unit, but some form of substitution takes place at the

aggregate level: as capital becomes cheaper and cheaper in efficiency units, the European

economy produces with more productive capital per worker and fewer workers.

27We reached this conclusion from their Table 1, where shocks are all bundled into a time effect. An idealcomparison with our model would be a measure of this institutional multiplier for their observable “labordemand” shock, but this number is not directly available in the paper.28In this experiment we set γ = 5% per year, the average of the period considered. All the other parameters

are unchanged.

41

Last but not least, our approach has the advantage over virtually all of the existing lit-

erature that formalizes the U.S.—Europe comparison of being able to measure the source of

the shock independently, through capital-embodied productivity. Thus we do not calibrate

the source of the shock to the labor market equilibrium outcomes of interest, such as wage

inequality or labor share.29 As a result, the focus of the analysis is limited to the different

unemployment experience of United States and Europe. We maintain the view that unem-

ployment, wage inequality and changes in the labor income share have been produced by

the same fundamental shock and should be explained jointly: they are all dimensions along

which the model needs to be evaluated rather than calibrated.

5.5 A comparison with the Aghion-Howitt/Mortensen-Pissaridessetup

In contrast to other search models with vintage capital (in particular, see Aghion and Howitt

1994, and Mortensen and Pissarides 1999) the pool of vacant firms in our model economies

is heterogenous. Vacancies are heterogeneous because of random matching and because the

bulk of the cost associated with creating a job is related to the purchase of the capital needed

in production. It is precisely this vacancy heterogeneity that contributes to the amplification

of the effects of shocks in our economies.

Traditional search-matching frameworks (e.g., Mortensen and Pissarides 1999) assume

that a new machine can be created at zero cost and that only posting a vacancy is costly.

This assumption implies that the pool of vacancies consists of the newest machines only,

and that only machines in existing matches age over time. In our setup we instead assume

that once a machine has been acquired at a cost, recruiting costs are “small compared to

29In Ljungqvist and Sargent (1998) the skill depreciation shock is calibrated to the increase in U.S. earn-ings instability. In Mortensen and Pissarides (1999a), the shock is a mean-preserving spread in the skilldistribution, calibrated to the increase in U.S. wage inequality. In Blanchard (1997), the main source of theshock is the degree of technological bias in favor of capital (against labor) and is identified through changesin the capital share of income. At the extreme of the spectrum the shock is completely unobservable. InMarimon and Zilibotti (1999) the shock is an increase in the degree of mismatch between workers and jobs,while Caballero and Hammour (1998) model an “appropriability” shock that changes the division of quasi-rents between capital and labor. Because of the intrinsic unobservability of the shock, no attempt is madeto calibrate the shock in these two papers.A recent exception is the paper by den Haan et al. (2001), where several candidates for the shock are

considered (the decline in total factor productivity and the rise in the real interest rate), all measured fromindependent data. However, the authors focus on unemployment and do not examine the dynamics of wageinequality or income shares.

42

that cost” (zero in our model).30 This explicit distinction between a “large” purchase/setup

cost for the machine — which is sunk when the vacant firm start searching— and a “smaller”

recruiting cost fits naturally with a vintage capital growth model, whose emphasis is on

capital investment expenditures as a way of improving productivity. Aghion and Howitt

(1994) also describe a vintage capital model with large setup costs for capital, but they

assume that matching is “deterministic”: at the time a new machine is set up, a worker

queues up for the machine, and after a fixed amount of time the worker and firm start

operations. Hence, in the matching process, all vacant firms are equal (although they do not

embody the leading-edge technology).

The equilibrium of the Mortensen-Pissarides (MP) model is characterized by the following

modified surplus, job creation, and job destruction equations:

eS(a; a) =

Z a

a

e−(r+δ)(a−a)(e−γa − eγ(a−a−a))da, (27)

e−γa = b+m(θ)β eS(0; a), (JC00)

I = λf (1− β) eS(0; a). (JD00)

The cost I now represents a flow search cost. Notice two differences between our setup and

the MP model. First, in the MP model old machines from an existing match are discarded

once the match dissolves. This means that there is no longer a non-zero outside option to the

firm, and the term (1− β)λf is eliminated from the effective discount factor in the surplus

function (27). Second, in the MP model workers always face the newest vintage in the pool

of vacancies. This means that in the job destruction condition (JD00), only the surplus of the

most recent vintage is relevant and not a weighted average of the surpluses across vintages.

The qualitative comparative statics of the MP model with respect to b and γ are the

same as in our model, but there are important quantitative differences. Compared to the

MP model, the just mentioned differences between (JC, JD) and (JC 00, JD00) dampen the

positive effect of a rise in γ on the right-hand sides of JC 00 and JD00 in our setup for

two reasons. First, as discussed in Section 4.1, the equilibrium density of vacancies shifts

towards older machines and for sufficiently old matches the surplus declines with the rate of

embodied technological change. Second, from (14), it is clear that the larger discount factor

weakens the effect of γ on the surplus. In conclusion, a given rise in γ increases the value

30At the risk of being redundant, let us restate that vacancy heterogeneity will survive the addition of aflow search cost c, as long as this cost is strictly less than the initial set-up cost I.

43

of vacant firms and the value of search in the standard model by a larger amount, which

implies ampler downward shifts of the two curves and a larger reduction of a. Thus, the

Mortensen-Pissarides economy responds to shocks mainly through the incidence margin and

not through the duration margin.

We now turn to wage inequality and the income shares. In the MP model the ratio of the

highest to the lowest wage is w(0)/w(a) = (1− β) + βeγa; thus, changes to wage inequality

induced by a rise in γ will only take place through the technological heterogeneity channel.

In relation to the labor share, since the Mortensen-Pissarides economy responds to the shock

mainly through the incidence margin, that is, through a large reduction in the destruction

age a rather than a rise in unemployment duration, it will have a smaller change in the labor

share, as explained in Section 4.3. Finally, note the absence of any effect due to the firms’

outside option both for inequality and for the labor share, contrary to what we found in

Section 4.2. These effects, which originate precisely from the vacancy heterogeneity, play an

important role in the quantitative analysis above.

Figure 7 shows that the MP model displays a weak interaction between changes in the

rate of embodied technological change γ and benefits, for empirically plausible values of b

(until b = 0.6).31 Unemployment duration barely responds to a change in the growth rate

γ, whereas the life-length of capital is reduced substantially, and in fact the separation rate

increases much more than in the baseline economy. The only case where an interaction is

evident is for b = 0.7, the extreme parametrization. However, the significant rise in unem-

ployment (4 points) is still well below its data counterpart. Moreover, as anticipated this

rise takes place along the “wrong” margin: it is unemployment incidence that increases by

8%, while duration goes up by a small amount (only 5 weeks). Wage inequality increases by

less than 0.5% in U.S.-type economies and is essentially constant in Europe-type economies.

In particular, the absence of the firm’s outside option effect prevents inequality from falling

in high b economies. Finally, the reduction in the labor share is insignificant, in both low-

and high-benefits economies.

31We have tried to calibrate the MP model to the same set of moments we selected for our economy inSection 5. In the process we have found that the MP model cannot match the average age of capital inthe data. It can generate at most an average age of capital equal to 7.5 years conditionally on matching allthe other moments. We let γ vary in the same range, while we change the values of b to generate similarbenefits-wage ratio as in the baseline experiment. The value b = 0.10 implies a benefits-wage ratio of 15%,b = 0.4 of 50%, b = 0.6 of 80% and b = 0.7 of 90%. The case b = 0.7 is not empirically plausible, but itis useful to interpret the results. The other model parameters are set as follows: r = 0.017, δ = 0.061, β =0.1, α = 0.55, I = 2.7, and A is normalized to 1.

44

3.5 4 4.5 5 5.5 6 6.53

5

7

9

11

13

15

17

19

21

γ

UNEMPLOYMENT RATE

b=.05b=.40b=.60b=.70

3.5 4 4.5 5 5.5 6 6.56

8

10

12

14

16

18

20

22

24

26

28

30

32

γ

UNEMPLOYMENT DURATION (weeks)

3.5 4 4.5 5 5.5 6 6.524

26

28

30

32

34

36

38

γ

ANNUAL SEPARATION RATE

3.5 4 4.5 5 5.5 6 6.52

2.5

3

3.5

4

4.5

5

5.5

6

6.5

7

7.5

8

γ

AVERAGE AGE OF CAPITAL (years)

3.5 4 4.5 5 5.5 6 6.50.01

0.02

0.03

0.04

0.05

0.06

γ

90-10 LOG WAGE DIFFERENTIAL

3.5 4 4.5 5 5.5 6 6.5

0.66

0.68

0.7

0.72

0.74

0.76

0.78

0.8

0.82

0.84

0.86

γ

LABOR SHARE

Figure 7: The effects of a rise in the rate of embodied technical change γ in the standardMortensen-Pissarides model, when the economies differ only via the level of welfare benefitsb.

45

6 Concluding remarks

The past twenty years have been marked by very rapid capital-embodied growth. In this

paper, we have made an attempt to understand how the benefits of this type of technological

change are shared among labor market players: who are the winners and the losers when

the relative productivity of new capital increases? In a generalized version of the standard

vintage capital model with labor market frictions, we showed qualitatively that the answer

depends crucially on the institutions and policies of the economy considered. With European-

style generous welfare benefits, faster technical change reduces the wage inequalities among

employed workers. It magnifies the differences between employed and unemployed workers,

as unemployment duration rises substantially with little change in job tenures, and it shrinks

the share of income going to labor earners, more strongly so where employment protection

legislation is particularly strict.

Quantitatively, the model suggests (i) that embodied technical change accounts for a small

but significant part of wage inequality, and (ii) that an acceleration of embodied technical

change together with different labor market policies can account for the differential labor

market outcomes in continental Europe and the United States. Given the absence of studies

on the impact of vintage effects on wage differentials, we have not calibrated the model

along this dimension, but rather we have used it to obtain an estimate of this elasticity. The

calibrated model suggests that embodied technical change accounts for one fourth of residual

wage inequality in the United States. The calibrated model is also successful in generating

the differential rise in unemployment between the United States and Europe following an

acceleration of technological change. When firing costs are embedded, the model can also

match the differential fall in the labor share, but it tends to overstate the fall in the U.S.

labor share. An acceleration in the rate of embodied technical progress leads to a small

increase in wage inequality in the U.S.-like economy and to a small fall in the Europe-like

economy with generous welfare benefits.

We conjecture that the two main shortcomings of the quantitative analysis (too low a rise

in inequality and too rapid a decline in the labor share in U.S.-type economies) result mainly

from the simplifying assumption that workers are ex ante equal. In Hornstein, Krusell, and

Violante (2002), we extend our environment to two skill levels. In particular, we investigate

whether such a model can generate much larger increases in wage inequality and a stable

46

labor share. Accounting for differences in ex ante labor quality might be important, because

in the United States the fall in the wage bill for unskilled workers seems to have been

accompanied by a rise in the demand for skilled workers and in their labor share.

Finally, we have compared our generalized version of the model to the standard version

with a trivial equilibrium distribution of vacancies to show that vacancy heterogeneity is an

important driving force behind the results. In the process, we realized that the standard

model predicts no change in labor shares. Although this is a shortcoming of the standard

model, recall that our model fails in the opposite direction since it generates too large a fall

in the U.S. labor share. A useful lesson that can be learned from the comparison of the two

models is that the larger is the initial set-up cost I, the steeper is the decline in the labor

share as γ rises. One could perhaps argue that certain policy differences between the U.S.

and Europe that affect I (e.g., red tape associated with starting new businesses) can also

help explaining the differential evolution of the labor shares.

47

Appendix

A.1 In the frictionless economy the competitive equilibrium allocation and the

Pareto-optimal allocations are the same.

The planner maximizes the discounted value of future consumption, i.e., output net of in-

vestment, subject to the constraint that the total number of machines in operation in each

period cannot exceed the aggregate labor force:

max{ef (t),a(t)}

Z ∞

0

e−rt(eγtZ a(t)

0

ef (t− a) e−γada− ef (t) Ieγt

)dt

s.t.

Z a(t)

0

ef(t− a)da ≤ 1, for all t.

The planner chooses the measure of entrants ef(t) of vintage t and the maximal age a (t) of

vintages which are operating at t. Since there are no frictions, the planner can use arbitrary

firm measures of any vintage, and since more recent vintages are more productive, the planner

uses all firms of the most recent vintages first. At time t the planner operates vintages in

[t− a (t) , t] . We can write the Lagrangian for the constrained optimization problem in

terms of the contributions of the different vintages as

L =

Z ∞

0

e−(r−γ)tef (t)

(Z a(t)

0

e−rada− I

)dt

−Z ∞

−a(0)ef (t)

(Z t+a(t)

t

ϕ (τ) e−(r−γ)τdτ

)dt+

Z ∞

0

e−(r−γ)tϕ (t) dt ,

where ϕ (t) is the Lagrange multiplier on the labor endowment constraint and a (t) ≡a [t− a (t)] denotes the age at which a vintage t machine is retired. The first order con-

ditions with respect to ef(t) and a(t) read, respectively,Z a(t)

0

e−rada− I −Z a(t)

0

e−(r−γ)aϕ(t+ a)da = 0,

e−ra(t)©1− eγa(t)ϕ [t+ a (t)]

ª= 0.

The first condition states that the planner will add new firms until the benefits (present value

of additional output) equal the direct installation costs and the indirect costs. This could

follow from the fact that the creation of new firms requires the destruction of others, given

the fixed amount of labor available. The second condition states that a marginal increase

48

in the destruction age raises the expected output of existing firms but once again requires a

reduction in the total number of operating firms.

In steady state, the time subscripts can be omitted and a = a. Moreover, we can impose

the condition ef = 1/a which guarantees that the distribution is stationary and all the labor

is employed. From the second condition, we obtain that ϕ = e−γa. This expression is easily

interpretable: ϕ is the multiplier on the total labor force constraint, and e−γa is exactly the

value of slackening this constraint, i.e., the marginal contribution of an extra unit of labor

(recall that this is also the equilibrium wage rate). Using this result in the first condition,

we arrive at

I =

Z a

0

e−ra£1− e−γ(a−a)

¤da

which is the key equilibrium condition (2) in the decentralized economy.

A.2 Derivations of value functions and employment distributions.

The value functions and distributions of our continuous-time model can be derived as limits

of a discrete time formulation. A typical derivation of the differential equations for value

functions (5)-(8) goes as follows. Consider the value of a vacant firm with capital of age a at

time t, V (t, a). For a Poisson matching process, the probability that the vacant firm meets

a worker over a small finite time interval [t, t+∆] is ∆λf . We can define the vacancy value

recursively as

V (t, a) = ∆λfhJ(t+∆, a+∆)− V (t+∆, a+∆)

i+ e−r∆V (t+∆, a+∆),

where the first term is the expected capital gain from becoming a matched firm with value

J and the second term is the present value of remaining vacant at the end of the time

interval. On a balanced growth path all value functions increase at the rate γ over time,

i.e., V (t, a) = eγtV (a) and J (t, a) = eγtJ (a). Subtracting V (t+∆, a) from both sides,

substituting the balanced growth path expressions for V and J , and dividing by ∆eγ(t+∆),

we can rearrange the value equation into

−e−γ∆V (a) eγ∆ − 1∆

= λf [J(a+∆)− V (a+∆)] +e−r∆ − 1

∆V (a+∆)

+V (a+∆)− V (a)

∆.

As we shorten the length of the time interval and take the limit for ∆ → 0, we obtain the

differential equation (5):

−γV (a) = λf [J(a)− V (a)]− rV (a) + V 0(a).

49

The equations describing employment dynamics are derived as follows. Consider the

measure of matched vintage a firms at time t. Over a short time interval of length ∆, the

approximate change in the measure is

µ(t+∆, a) = µ(t, a−∆)(1−∆δ) +∆λfν(t, a−∆).

Subtracting µ (t, a) from both sides and dividing by ∆ we obtain

µ(t+∆, a)− µ(t, a)

∆= −µ(t, a)− µ(t, a−∆)

∆− δµ(t, a−∆) + λfν(t, a−∆).

Taking the limit for ∆→ 0 one obtains

µt(t, a) = −µa(t, a)− δµ(t, a) + λfν(t, a).

At steady state, these measures do not change with t, and we obtain the result stated in

(16).

Similarly, the differential equation for unemployment can be derived as follows. Over a

short time period of length ∆ the change in unemployment is

u(t+∆) = u(t)

·1−

Z a

0

∆λw(a)da

¸+∆δ

Z a

0

µ(t, a)da+

Z ∆

0

µ(t, a− x)dx

The first two terms on the right-hand side are standard: they are flows assuming a Pois-

son process and these flows are approximately linear in the length of the interval, since

the interval is small. The third term sums all those matches that will reach a by the end

of the period and therefore separate. Subtracting u(t) on both sides, dividing by ∆, tak-

ing limits as ∆ approaches 0, and assuming steady state yields the result (21). To find

lim∆→0hR ∆

0µ(a− x)dx

i/∆, use l’Hopital’s rule.

Given the differential equation for employment (16), we can easily determine that

µ(a) =λfν(0)

δ + λf

£1− e−(δ+λf )a

¤and that (28)

ν(a) =ν(0)

δ + λf

£δ + λfe

−(δ+λf )a¤ . (29)

Thus, the total number of vacancies, v, satisfies

v =

Z a

0

ν(a)da =ν(0)

δ + λf

½aδ +

λfδ + λf

£1− e−(δ+λf )a

¤¾. (30)

50

Integrating both sides of the equation ν(a)+µ(a) = ν(0) over the support [0, a), we conclude

that the total number of matched pairs (employment), µ, satisfies µ = ν(0)a− v, or

µ =ν(0)λfδ + λf

½a− 1

δ + λf

£1− e−(δ+λf )a

¤¾. (31)

Solving (21) for u, and substituting in for µ(a)/µ, we arrive at

u =1 + δ

³a

1−e−(δ+λf )a −1

δ+λf

´1 + [δ +m(θ, 1)]

³a

1−e−(δ+λf )a −1

δ+λf

´ . (32)

Having found the unemployment rate u, the entry of firms ν(0) is simply found from equation

(31), using the fact that µ = 1−u. Equations (17) and (18) in the main text can be derived

simply using (28) together with (31) and (29) together with (30), respectively.

A.3 Proof of Lemmas 1,2,3 (the job creation curve).

Lemma 1 (the downward sloping (JC) curve): The (JC) curve is implicitly defined by

the equation

I = (1− β)λf

Z a

0

e−(r−γ)aS (a; a, λf , γ) da. (JC)

We show that the RHS of this expression is increasing in a and λf , which implies that the

(JC) curve is downward sloping in the (λf , a) space.

Straightforward integration of the equation (14) defining the surplus equation yields

S (a; a, λf) = e−γa¡1− e−σ0(a−a)

¢/σ0 − e−γa

¡1− e−σ1(a−a)

¢/σ1 (33)

with σ0 = r + δ + (1− β)λf and σ1 = σ0 − γ. It is immediate that the surplus function is

decreasing in a and increasing in a. Since the surplus function is increasing in the exit age

a, it is immediate that the RHS of (JC) is increasing in the exit age a.

To show that the RHS of (JC) is increasing in λf , rewrite the integral as

I = e−γaZ a

0

e−(r−γ)a½Z a−a

0

λfe−(ρ+λf)a £eγ(a−a−a) − 1¤ da¾ da (34)

with ρ = r − γ + δ and λf = (1− β)λf . We now show that the integral of the function

f (a;λf) = λfe−(ρ+λf)a with respect to the weighting function g (a) = eγ(a−a−a)−1 is increas-

ing in λf . The function f is increasing (decreasing) with respect to λf for a < (>) a = 1/λf .

51

The integral of the function f , however, is increasing with λf , asZ a

0

f (a;λf) da =h1− e−(ρ+λf)a

i λfρ+ λf

∂

∂λf

Z a

0

f (a;λf) da =ρ

ρ+ λf

Z a

0

f (a;λf) da+λf

ρ+ λfe−(ρ+λf)a > 0.

The integral of f with respect to g is also increasing with λf , since the weighting function g

is monotonically decreasing in a,

∂

∂λf

Z a−a

0

f (a;λf) g (a) da

=

Z a+

0

fλf (a;λf) g (a) da+

Z a−a

a+

fλf (a;λf) g (a) da

>

Z a+

0

fλf (a;λf) g (a+) da+

Z a−a

a+

fλf (a;λf) g (a+) da

= g (a+)

Z a−a

0

fλf (a;λf) da > 0,

with a+ = min {a, a− a}.Lemmas 2 and 3 (the asymptotes of the (JC) curve): Integrating equation (JC)

yields

I =(1− β)λf

r + δ + (1− β)λf(35)(

1− e−ra

r− σ0

σ1e−γa

1− e−(r−γ)a

r − γ+ e−ra

1− e−(δ+(1−β)λf)

δ + (1− β)λf

γ

σ1

).

Taking the limit of expression (35) as λf →∞, we get

I =1− e−ramin

r− e−γamin

1− e−(r−γ)amin

r − γ

=

Z amin

0

e−ra£1− e−γ(amin−a)

¤⇒ amin = aCE,

where aCE is the age cut-off of the frictionless economy, implicitly defined by (2). Alterna-

tively taking the limit of expression (35) as a→∞, we get

I =(1− β)λminf

r + δ + (1− β)λminf

1

r⇒ λminf =

rI

1− rI

r + δ

1− β.

A.4 Proof of Lemmas 4 and 5 (the job destruction curve).

52

Lemma 4 (the upward-sloping (JD) curve): The (JD) curve is implicitly defined by

the equation

1 = beγa + β

Z a

0

λw(a; a, λf)eγaS(a; a, λf , γ)da. (JD)

We show that the RHS of this expression is increasing in a and decreasing in λf , which

implies that the (JD) curve is upward-sloping in (λf , a) space.

(4a) The RHS of (JD) is increasing in a: The first term is clearly increasing in a. Now

take the derivative of the function to be integrated in the second term, and express it in

terms of elasticities (∂λw∂a

a

λw+

∂S

∂a

a

S

)Sλwa

(36)

where S = eγaS. The elasticity of the density λw is given by

∂λw∂a

a

λw= − δa+ λf ae

−(δ+λf)a

δa+ λfR a0e−(δ+λf)ada

.

Thus the first term in (36) is negative, but its absolute value is less than one since e−(δ+λf)a >

e−(δ+λf)a for a ≤ a. We will now show that the elasticity of the modified surplus function S

with respect to a is positive and greater than or equal to one. It will therefore follow that

the integral in (JD) is increasing in a.

We proceed in three steps. First, we show that the elasticity of S with respect to a is

increasing in a for a given a. That is, if the elasticity is greater than one at a = 0, then it is

greater than one for all a. Second, we show that for small enough a the elasticity is greater

or equal to zero at a = 0; in particular, we show that lima→0³∂S/∂a

´³a/S

´≥ 1. Third,

we show at a = 0 the elasticity is increasing in a. The three steps together imply that the

elasticity is greater or equal to one for all a ≤ a.

The elasticity of S with respect to a is given by

∂S

∂a

a

S=

γa

1−H (a− a)with H (x) ≡ e−γx

(1− e−σ1x) /σ1(1− e−σ0x) /σ0

.

The sign of the derivative of the function H is given by

sign (H 0) = σ0γe−(σ0+γ)

½Z x

0

eγydy −Z x

0

eσ0ydy

¾.

Since σ0 = r + δ + (1− β)λf and by assumption r > γ, the function H is decreasing in x.

Therefore the elasticity is increasing in a.

53

The limit of the elasticity at a = 0 as a converges to zero is greater or equal to one.

To see this note that for a → 0, the numerator and denominator converge to zero, and by

l’Hopital’s rule

lima→0

a∂S/∂a

S= lim

a→0

a³∂2S/∂a2

´+ ∂S/∂a

∂S/∂a= 1 + lim

a→0

³∂2S/∂a2

´a

∂S/∂a≥ 1

since ∂S/∂a, ∂2S/∂a2 ≥ 0.Finally we need to show that at a = 0, the elasticity is increasing in a, that is G (a) =

a/ [1−H (a)] is increasing in a. First, multiply numerator and denominator by σ1(1−e−σ0a).This delivers

G (a) =σ1a(1− e−σ0a)

σ1(1− e−σ0a)− σ0(1− e−σ1a)e−γa= a

σ1(1− e−σ0a)σ1 − σ0e−γa + γe−σ0a

.

Notice that the denominator of this expression is positive: at a = 0, it equals 0, and

its derivative equals γσ0(e−γa − e−σ0a), which is positive because of the assumption that

γ < r < σ0. For large a, the expression is large: lima→∞G (a) = lima→∞ a =∞.The derivative of G equals

G0 (a)σ1

=1− e−σ0a

σ1 − σ0e−γa + γe−σ0a

−aσ0e−σ0a (σ1 − σ0e

−γa + γe−σ0a)− (1− e−σ0a) γσ0 (e−γa − e−σ0a)

(σ1 − σ0e−γa + γe−σ0a)2

=1− e−σ0a

σ1 − σ0e−γa + γe−σ0a

−aσ0σ1e

−σ0a − σ0e−(γ+σ0)a + γ

¡e−2σ0a − e−γa + e−σ0a + e−(γ+σ0)a − e−2σ0a

¢(σ1 − σ0e−γa + γe−σ0a)2

.

Substituting for σ1 = σ0 − γ and simplifying yields

G0 (a)σ1

=1− e−σ0a

σ1 − σ0e−γa + γe−σ0a− aσ0

σ0¡e−σ0a − e−(γ+σ0)a

¢+ γ

¡e−(γ+σ0)a − e−γa

¢(σ1 − σ0e−γa + γe−σ0a)2

.

Thus, it is sufficient to study¡1− e−σ0a

¢ ¡σ1 − σ0e

−γa + γe−σ0a¢− aσ0

£σ0¡e−σ0a − e−(γ+σ0)a

¢+ γ

¡e−(γ+σ0)a − e−γa

¢¤.

This expression equals¡1− e−σ0a

¢ ¡σ1 − σ0e

−γa + γe−σ0a¢− aσ0

£σ0e

−σ0a ¡1− e−γa¢− γe−γa

¡1− e−σ0a

¢¤.

54

Factorizing, we are left with¡1− e−σ0a

¢µσ1 − σ0e

−γa + γe−σ0a − aσ20e−σ0a 1− e−γa

1− e−σ0a+ aγσ0e

−γa¶.

The left factor is larger than zero (it starts at zero and increases). After substituting for

σ1 = σ0 − γ, the right factor can be rewritten as

σ0¡1− e−γa

¢− γ¡1− e−σ0a

¢+ σ0γa

µe−γa − e−σ0a

1− e−γa

1− e−σ0aσ0γ

¶.

Utilizing a simple integral formula, this becomes

σ0γ

µZ a

0

e−γxdx−Z a

0

e−σ0xdx¶+ σ0γa

Ãe−γa − e−σ0a

R a0e−γxdxR a

0e−σ0xdx

!.

The first of these two terms is positive, since σ0 > γ by assumption. If we can show that

the second term is positive, we are done. That term has two sub-terms; we will prove that

their ratio exceeds one. The ratio reads

e−γaR a0e−σ0xdx

e−σ0aR a0e−γxdx

=

R a0e−(γ+σ0)xe−(a−x)γdxR a

0e−(γ+σ0)xe−(a−x)σ0dx

.

But since the weighting function e−(a−x)γ is everywhere above the weighting function e−(a−x)σ0,

again because σ0 > γ (and a > x), and the rest of the integrand is positive, the ratio indeed

must exceed 1.

(4b) The RHS of (JD) is decreasing in λf for a Cobb-Douglas matching function

with α > 1/2: We rewrite equation (JD) as

1 = beγa + β [m (θ, 1)λf ]

Z a

0

·λw(a; a, λf)/λf

m (θ, 1)

¸S(a; a, λf , γ)da.

It is immediate that the two terms under the integral, the modified density λw and the

modified surplus function S, are decreasing in λf . Assuming a Cobb-Douglas matching

function and substituting for θ, the term pre-multiplying the integral becomes

Aλ(1−2α)/(1−α)f

which is decreasing in λf for α > 1/2.

Lemma 5 (The asymptote of the (JD) curve): For λf large, the density λw converges

to a uniform density on [0, a], limλf→∞ λw (a) =1

a+ 1/δ, and the surplus function converges

to zero, limλf→∞ S (a;λf , γ) = 0. Therefore equation (JD) converges to

1 = beγamax ⇒ amax = − ln (b) /γ.

55

A.5 Proof of Lemma 6,7,8, and 9 (comparative statics).

Lemma 6 (b): Obvious from inspection of (JC) and (JD).

Lemma 7 (γ): The RHS of (JC) is increasing in the rate of embodied technical change γ,

because the function f (γ) = eγaS (a; a, λf , γ) is increasing in γ. The derivative of f with

respect to γ is∂f

∂γ= e−γx

¡σ1x+ e−σ1x − 1¢ /σ21

with x = a− a. Notice that ∂f/∂γ = 0 at x = 0, and that ∂f/∂γ is increasing in x

∂2f

∂γ∂x= σ1

¡1− e−σ1x

¢ ≥ 0 for x ≥ 0.Since the RHS of (JC) is increasing γ and a, the (JC) curve shifts downward as γ increases.

The RHS of (JD) is increasing in the rate of embodied technical change γ because the

function S ≡ eγaS is increasing in γ. The derivative of S with respect to γ is

∂S

∂γ= xeγx

¡1− e−σ0x

¢/σ0 +

£e−σ1x (1 + σ1x)− 1

¤/σ21

with x = a− a. Notice that ∂S/∂γ = 0 at x = 0 and that ∂S/∂γ is increasing in x, i.e.,

∂2S

∂γ∂x= eγx (1 + γx)

¡1− e−σ0x

¢/σ0 ≥ 0 for x ≥ 0.

Therefore ∂S/∂γ ≥ 0 for all x ≥ 0. Since S is increasing in γ for all a ∈ [0, a], the RHS of(JD) is increasing in γ. Since the RHS of (JD) is increasing in γ and a, the (JD) curve shifts

downward as γ increases.

Lemma 8 (r): Obvious from inspection of (JC) and (JD).

Lemma 9 (A): When the frictions disappear, both meeting probabilities tend to infinity.

We proved above that as λf → ∞, the job creation condition converges to the competitivecondition (2). Consider now the wage equation (23). It is easy to compute that as λf →∞,the term λf (1− β)S(a) converges to e−γa−e−γa, implying w(a) = (1−β)(r−γ)U+βe−γa =e−γa, where the last equality follows from condition (13). Thus, the job destruction condition

converges to the competitive one as well, which proves the Lemma.

A.6 Solution of the model with the firing tax.

56

The value equations in the model with the firing tax/hiring subsidy are:

(r − γ)V (a) = max{λf [T + J(a)− V (a)] + V 0(a), 0}(r − γ)J(a) = max{e−γa − w(a)− δ [J(a)− V (a) + T ] + J 0(a), (r − γ)[V (a)− T ]}(r − γ)U = b+

Z a

0

λw(a) [W (a)− U ] da

(r − γ)W (a) = max{w(a)− δ [W (a)− U ] +W 0(a), (r − γ)U},

where it is clear that the firm receives from the government the subsidy T upon hiring and

pays a tax of the same amount T upon separation. The total surplus S (a) of a type a match

is now S (a) ≡ J (a) +W (a) − V (a) − U + T both for a new meeting and for an ongoing

relationship. The term T plays the role of the subsidy in the first case, and the role of the

tax in the second case (with the subsidy being sunk): we can avoid the two-tier structure.

The surplus is split according to the Nash rule β[W (a)−U ] = (1− β)[T + J(a)− V (a)] and

using this rule with the definition of the surplus, we obtain a differential equation for the

surplus with solution

S(a; a, λf) =

Z a

a

e−[r−γ+δ+λf (1−β)](a−a)£e−γa − (r − γ)(U − T )

¤da, (37)

where the associated destruction rule for a match is (25) in the main text. Using the value

equation for the employed worker and the Nash rule, we arrive at the expression for the

wage function (26) in the main text. Using the Nash splitting rule into the equation for the

value of a vacancy and solving the associated differential equation yields exactly the same

job creation condition as in the benchmark model with the implication that the destruction

rule for a vacancy V (a) = 0 implies S(a; a, λf) = 0; thus, a = a. Using the value equation for

the unemployed worker into the destruction condition (25) yields the equilibrium condition

(JD’) in the main text. Finally, it is easy to see that the expressions for all the distributions

are unchanged.

57

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61

Table 1: Cross-country labor market data

1965 1970 1975 1980 1985 1990 1995 Change

Unemp. Rate 0.018 0.011 0.017 0.029 0.045 0.054 0.061 0.043

Austria Labor share 0.698 0.679 0.717 0.694 0.665 0.646 0.645 -0.053Inequality 0.820 0.790 0.870 0.880 0.060

Unemp. Rate 0.023 0.022 0.064 0.114 0.111 0.110 0.142 0.120

Belgium Labor share 0.667 0.729 0.730 0.682 0.685 0.676 0.009

Inequality 0.660 0.650 0.640 -0.020

Unemp. Rate 0.014 0.016 0.061 0.093 0.085 0.112 0.103 0.089

Denmark Labor share 0.736 0.723 0.732 0.706 0.677 0.635 0.605 -0.131

Inequality 0.760 0.770 0.770 0.010

Unemp. Rate 0.025 0.021 0.050 0.051 0.047 0.121 0.167 0.142

Finland Labor share 0.738 0.711 0.762 0.730 0.723 0.733 0.680 -0.058

Inequality 0.890 0.920 0.940 0.930 0.040

Unemp. Rate 0.020 0.027 0.049 0.079 0.101 0.105 0.115 0.095

France Labor share 0.688 0.674 0.707 0.710 0.645 0.618 0.603 -0.085

Inequality 1.210 1.210 1.240 1.230 0.020

Unemp. Rate 0.010 0.011 0.037 0.060 0.075 0.078 0.099 0.089

Germany Labor share 0.685 0.703 0.703 0.704 0.667 0.658 0.637 -0.048

Inequality 0.870 0.830 0.830 0.810 -0.060

Unemp. Rate 0.019 0.025 0.044 0.089 0.091 0.086 0.079 0.060

Greece Labor share 0.693 0.699 0.698 0.694 0.690 0.712 0.692 -0.002

Inequality

Unemp. Rate 0.047 0.055 0.078 0.112 0.164 0.146 0.120 0.073

Ireland Labor share 0.828 0.842 0.835 0.833 0.763 0.715 0.645 -0.183

Inequality

Unemp. Rate 0.041 0.043 0.051 0.070 0.099 0.096 0.120 0.079

Italy Labor share 0.669 0.687 0.711 0.690 0.656 0.653 0.606 -0.063

Inequality 0.850 0.830 0.770 0.970 0.120

Unemp. Rate 0.010 0.018 0.038 0.080 0.081 0.062 0.071 0.061

Netherlands Labor share 0.656 0.687 0.705 0.661 0.623 0.619 0.624 -0.032

Inequality 0.920 0.960 0.950 0.030

Unemp. Rate 0.016 0.015 0.018 0.026 0.030 0.056 0.049 0.034

Norway Labor share 0.750 0.771 0.782 0.757 0.739 0.713 -0.037

Inequality 0.720 0.720 0.680 -0.040

Unemp. Rate 0.040 0.024 0.065 0.079 0.070 0.051 0.073 0.033

Portugal Labor share 0.562 0.615 0.873 0.751 0.673 0.679 0.680 0.118

Inequality

Unemp. Rate 0.028 0.030 0.059 0.161 0.200 0.196 0.230 0.202

Spain Labor share 0.763 0.780 0.788 0.756 0.679 0.669 0.616 -0.147

Inequality

Unemp. Rate 0.018 0.022 0.019 0.028 0.021 0.052 0.079 0.061

Sweden Labor share 0.724 0.716 0.745 0.711 0.691 0.693 0.630 -0.095

Inequality 0.750 0.760 0.730 0.790 0.040

Unemp. Rate 0.019 0.025 0.044 0.089 0.091 0.086 0.079 0.060

UK Labor share 0.693 0.699 0.698 0.694 0.690 0.712 0.692 -0.002

Inequality 0.920 1.050 1.150 1.200 0.280

Unemp. Rate 0.040 0.058 0.076 0.099 0.089 0.103 0.096 0.056

Canada Labor share 0.716 0.660 0.652 0.634 0.630 0.666 0.659 -0.057

Inequality 1.240 1.390 1.380 1.330 0.090

Unemp. Rate 0.038 0.054 0.070 0.083 0.062 0.066 0.055 0.017

USA Labor share 0.685 0.695 0.675 0.678 0.665 0.666 0.670 -0.015

Inequality 1.180 1.350 1.380 1.470 0.290

Europe Unemp. Rate 0.023 0.024 0.046 0.077 0.087 0.095 0.108 0.084

Average Labor share 0.707 0.711 0.749 0.723 0.684 0.673 0.641 -0.058

Inequality 0.859 0.841 0.844 0.900 0.040

Note: Data on unemployment rates are from Blanchard and Wolfers (2000). Data on labor shares are from Blanchard and Wolfers (2000) except the 1995 entry for Austria, Denmark, Ireland and Portugal which was computed directly from OECD data. Inequality is measured as the 90-10 log-wage differential for male workers. The data are taken from the OECD Employment Outlook (1996, Table 3.1). Austria: the measure is the 80-10 differential and data in the 1985 column are for 1987. Belgium: the measure is the 80-10 differential and data in the 1995 column are for 1993. Denmark: 1985 and 1990 columns are for 1983 and 1991 respectively. Finland: data in the 1985 column are for 1986. Germany: data in the 1985 and 1995 columns are for 1983 and 1993 respectively. Italy: data in the 1985, 1990 and 1995 columns are for 1984, 1991 and 1993 respectively. Netherlands: the measure of inequality is for males and females. Norway: data in the 1985 and 1990 columns are for 1983 and 1991 respectively. Moreover, the measure of inequality is for males and females. Portugal: data in the 1990 and 1995 columns are for 1989 and 1993 respectively. Canada: data in the 1980 and 1985 columns are for 1981 and 1986 respectively. For all countries, except US and UK, data in the 1995 column are for 1994. Europe average: unweighted mean of European countries, except UK.